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EMPIRICAL    STUDIES   IN   THE 
THEORY   OF   MEASUREMENT 


BY 

EDWARD  L.  THOKNDIKE, 

Professor  of  Educational  Psychology  in  Teachers  College,  Columbia  University. 


ARCHIVES  OF     PSYCHOLOGY 
EDITED   BY   R.  S.  WOODWORTH 


No.  3,  APRIL,  19O7 


Columbia  University  Contributions  to  Philosophy  and  PsychologyrVol.  XV.     No.  3 


ORK 
THE   SCIENCE 


•e 


CONTENTS 


MEASUREMENTS  OF  TYPE  AND  VARIABILITY 

§  1.     The  Comparative  Accuracy  of  the  Average  and  the  Median 1 

§  2.  The  Comparative  Accuracy  of  the  Mean  Square  Deviation  and  the 

Average  Deviation  5 

§  3.  The  Divergencies  of  the  Obtained  from  the  True  Measures  by 

Theory  and  by  Experiment  8 

§  4.  The  Relation  between  the  Amount  of  a  Central  Tendency  and 
the  Amount  of  the  Variability  of  the  Group  about  the  Cen- 
tral Tendency 9 

MEASUREMENTS  OF  RELATIONSHIPS 

§  5.     The  Meaning  of  Typical  Measures  of  Relationship 15 

§  6.     The  Presuppositions  of  Measures  of  Relationship   25 

§  7.     The  Advantages  of  the  Different  Measures    25 

§  8.     The  Attenuation  of  Measurements  of  Relationship   35 

§  9.     Minor  Advice  to  Students  of  Mental  and  Social  Relationships  ...  41 


EMPIRICAL  STUDIES   IN   THE  THEORY   OF 
MEASUREMENT 

IN  the  present  condition  of  psychology,  sociology  and  education, 
convenience,  economy  and  directness  are  as  important  desiderata  in 
methods  of  measurement  as  refinement  with  respect  to  precision. 
The  results  of  these  studies  justify  certain  methods  which  have  the 
decided  advantage  of  giving  measures  which  are  direct  functions  of 
the  data,  independent  of  any  hypothesis  about  the  prevalence  of  the 
so-called  'normal'  distribution,  but  which  have  been  somewhat  dis- 
countenanced or  at  least  neglected  in  both  the  theory  and  the  prac- 
tice of  statistics. 

The  section  on  correlation  attempts  also  to  make  clear  just  what 
is  measured  by  a  coefficient  of  correlation  and  what  the  dangers  are 
in  the  application  of  correlation  formulas  without  constant  super- 
vision by  an  adequate  sense  for  the  concrete  individual  facts  to  be 
related. 

MEASUREMENTS  OF  TYPE  AND  VARIABILITY 

§  1.     The  Comparative  Accuracy  of  the  Average  and  the  Median 

The  median  as  a  measure  of  the  central  tendency  of  a  series  of 
measures"~has  the  advantages  of  greater  quickness  of  calculation, 
freedom  from  the  influence  of  erroneous  measurements,  ease  of  in- 
terpretation and  often  greater  practical  significance.  It  is,  there- 
fore, important  to  know  whether  the  accuracy,  with  which  the 
median  actually  obtained  from  a  small  sampling  of  a  series  conforms 
to  the  true  median  of  the  total  series,  is  much  less  than  the  similar 
accuracy  in  the  case  of  the  more  commonly  used  measure,  the  average. 

It  is  possible  with  any  given  form  of  distribution  to  calculate  on 
the  basis  of  the  theory  of  probability  the  accuracy  in  either  case. 
Trusting  that  some  one  will  soon  do  this  for  typical  forms  of  dis- 
tribution other  than  the  so-called  'normal'  I  have  chosen  to  get 
empirical  data  on  the  same  question  from  actual  experiments  with 
random  samplings  from  certain  large  series  of  measures. 

The  median  was  calculated  for  each  sampling  by  regarding  the 
total  series  as  measures  of  a  continuous  variable,  quantity  61,  for 
instance,  equalling  from  60.0  up  to  62.0,  quantity  63  equalling  from 
62.0  up  to  64.0,  etc.  Where  the  median  fell  within  a  unit  of  the 
scale,  as  of  course  it  usually  did,  the  fractional  part  was  taken 

1 


2  EMPIRICAL    STUDIES    OF    MEASUREMENT 

which  would  be  correct,  supposing  the  cases  within  that  unit  of  the 
scale  to  be  equally  frequent  in  all  equal  subdivisions  of  that  unit 
of  scale. 

The  series  used  were  the  four  presented  in  Table  I.  A  is  an 
almost  perfect  representative  of  the  so-called  'normal'  surface  of 
frequency,  limited  at  about  -f  3.2*  and  —  3.2*.  B  is  also  a  sym- 
metrical distribution  following,  but  not  so  closely,  the  so-called  'nor- 
mal' type.  C  is  a  skewed  distribution  of  the  kind  so  frequently 
found  in  mental  and  social  measurements.  D  is  a  flattened  and 
rather  sharply  cut-off  type  of  distribution,  such  as  occurs  often  in 
facts  subject  to  conventional  regulation.  The  number  of  cases  was 
for  A  1,000,  for  B  1,307,  for  C  1,250  and  for  D  600.  The  mechan- 
ical arrangement  of  each  series  was  simply  so  many  small  cards  or 
slips  of  paper  each  with  a  number  written  on  it.  In  each  series 
these  cards  were  approximately  of  the  same  size,  shape  and  weight. 
From  such  a  series,  properly  shuffled  in  a  large  bowl,  drawings  were 
made. 

The  total  number  of  cases  in  any  series  is  of  course  of  no  signifi- 
cance. Whether  a  series  contains  1,000,  1,100,  1,426,  13,982  or 
160,000  cases  makes  no  appreciable  difference  to  any  of  the  matters 
to  be  investigated  here,  and  in  the  case  of  a  distribution  of  the  type 
of  D,  drawings  of  100  from  6,000  cases  would  not  differ  appre- 
ciably from  drawings  from  600.  The  reason  for  the  particular 
sizes  of  the  total  series  was  economy  of  time. 

It  is  most  convenient  to  arrange  series  for  such  experiments 
with  measures  •+•  and  —  from  the  central  tendency,  as  in  B  and  D ; 
the  time  of  recording  the  results  of  draws  is  lessened  and  also  the 
likelihood  of  errors.  Thus  in  A  —31,  —37,  —35,  etc.,  would 
be  better  than  61,  63,  65,  etc.  I  give  the  series,  however,  in  just 
the  way  they  were  made  and  used. 

Every  drawing  of  10  or  50  or  55  or  whatever  number  of  cases 
was  made  from  the  full  series.  However,  a  draw  of  10  having  been 
made  and  recorded,  a  draw  of  50  was  obtained  by  adding  40  to  the 
10  and  one  of  100  by  adding  50  to  the  50.  The  100  is  thus  from 
the  full  series,  but  is  obtained  with  a  saving  of  time. 

As  a  rule  drawings  of  10  or  11,  50  or  55,  100  or  110  and  275 
were  made,  but,  with  the  larger  drawings,  if  not  exactly  50  or  100 
were  drawn,  the  drawing  was  still  utilized.  Of  course  exact  sim- 
ilarity in  the  size  of  the  drawings  is  of  no  consequence  whatever  to 
any  of  the  conclusions  drawn. 


MEASUREMENTS    OF    TYPE    AND    VARIABILITY 
TABLE    I. 


Quan- 
tity 

A 

Fre- 
quency 

Quan- 
tity 

B 
Fre- 
quency 

Quan- 
tity 

C 
Fre- 
quency 

D 
Quan-     Fre- 
tity    quency 

61 

1 

1 

30 

—  7      20 

3 

1 

5 

1 

2 

80 

—  5     80 

7 

2 

9 

3 

3 

140 

—  3     100 

71 

5 

3 

6 

—  27 

1 

4 

175 

—  1     100 

5 

9 

—  25 

7 

12 

—  23 

2 

5 

200 

+  1     110 

9 

15 

—  21 

2 

81 

20 

—  19 

8 

6 

160 

+  3     90 

3 

26 

—  17 

10 

5 

31 

—  15 

26 

7 

120 

+  5     70 

7 

37 

—  13 

28 

9 

43 

—  11 

58 

8 

95 

+  7     30 

91 

50 

—  9 

62 

3 

54 

—  7 

98 

9 

80 

5 

59 

—  5 

102 

7 

62 

—  3 

128 

10 

60 

9 

63 

—  1 

129 

101 

63 

.+  1 

132 

11 

45 

3 

62 

+  3 

125 

5 

59 

+  5 

102 

12 

35 

7 

54 

+  7 

98 

9 

50 

+  9 

64 

13 

20 

111 

43 

+  11 

56 

3 

37 

+  13 

28 

14 

10 

5 

31 

+  15 

26 

7 

26 

+  17 

11 

9 

20 

+  19 

7 

121 

15 

+  21 

2 

3 

12 

+  23 

1 

5 

9 

+  25 

1 

7 

6 

+  27 

9 

5 

131 

3 

3 

2 

5 

1 

7 

1 

9 

1 

Av.         100                                       0                             6.0  .0 

Med.      100                                       0                             5.5  .0 

A.D.       10.0                                    6.2                             2.3  3.13 

a             12.4                                      7.8                              2.9  3.68 

Q.             8.4                                      5.4                              2.0  2.94 

A.D.  =  the  average  deviation  from  the  average. 
a       :=zthe  mean  square  deviation  from  the  average. 

Q.     =  one  half  the  difference  between  the  25  percentile  and  75  percentile 
measures. 


4  EMPIRICAL    STUDIES    OF    MEASUREMENT 

The  results  of  these  drawings  are  summarized  in  Table  II.  In 
Table  II.,  Nt  =  ihe  number  of  sets  drawn;  ATc  =  the  number  of 
cases  in  each  set;  Av.=the  average  divergence  of  the  obtained1 
from  the  true2  average;  Med.  =  the  average  divergence  of  the  ob- 
tained from  the  true  median;  A.D.  =  the  average  divergence  of 
the  obtained  from  the  true  average  deviation;  <r  =  the  average 
divergence  of  the  obtained  from  the  true  mean  square  deviation; 
Q.  =  the  average  divergence  of  the  obtained  from  the  true  (75  per- 
centile  —  25  percentile)/2. 

The  figures  for  the  last  three  divergences  under  'Actual'  are  the 
direct  results;  the  figures  under  'Percentile'  are  these  divergences 
in  percents  of  the  true  fact. 

The  table  shows  that  there  is  not  enough  superiority  in  accuracy 
in  any  case  to  outweigh  the  practical  advantages  which  the  median 
has  as  a  measure  of  such  quantities  as  prevail  in  the  mental  sciences.3 
The  divergences  of  the  medians  are  on  the  whole  only  about  22  per 
cent,  greater  than  those  of  the  averages,  f^ 

TABLE    II. 
DIVERGENCE  OF  OBTAINED  FROM  TRUE 

ACTUAL  PERCENTILE 

Nt      Nc.  Av.     Med.  A.D.          <r  Q.  A.D.          a          Q. 

SERIES 
A 


3 

102 

.47 

.33 

.33 

.53 

.77 

.33 

.43 

.92 

4 

53 

.92 

1.07 

.78 

1.08 

.80 

.78 

.87 

.95 

10 

10 

3.04 

3.60 

1.13 

1.59 

1.22 

1.13 

1.28 

1.45 

SERIES 

B 

3 

100 

.57 

.47 

.27 

.37 

.13 

.44 

.47 

.24 

4 

50 

1.25 

1.16 

.88 

.65 

1.05 

1.42 

.83 

1.94 

10  1.80     2.14  1.28     1.50       .70  2.06     1.92     1.30 


2  275  .045  .055  .05  .075  .055  .22  .26  .28 

3  110  .11  .17  .04  .04  .21  .17  .14  1.05 
5       55      .  .33  .43  .18  .18  .22  .78  .62  1.10 

25       11--'  .62  .77 


,.  7      100 


.33       .53  .097     .081     .14  .31       .22       .48 

T.O  .34       .59  .19       .16       .21  .61       .44       .71 

.86     1.04 

1  Obtained,  that  is,  from  the  limited  number  of  cases  in  the  drawing  in 
question. 

2 '  True '  meaning  here  that  obtainable  if  all  the  measures  of  the  total 
series  are  taken. 

8 The  general  merits  of  the  median  are  not  discussed  in  this  report;  so 
also  in  the  case  of  the  general  merits  of  the  average  deviation  and  of  per- 
centile  measures  of  variability.  To  thoughtful  students  of  mental  measure- 
ments they  will  be  obvious.  The  matter  is  briefly  discussed  in  my  '  Mental 
and  Social  Measurements'  ('04)  pp.  37  ff.  and  passim. 


MEASUREMENTS    OF    TYPE    ASD    VARIABILITY  5 

§2.     The   Comparative  Accuracy  of  the  Mean  Square  Deviation 

(called  by  various  authors  <r,  /*,  c,  S.D.,  or  Standard 

Deviation)  and  the  Average  Deviation 

The  most  burdensome  of  the  ordinary  statistical  operations  is 
the  calculation  of  the  square  root  of  the  average  of  the  squares  of 
the  deviations  of  a  series  of  measures  from  their  central  tendency, 
that  is,  of  the  mean  square  deviation.  In  the  case  of  the  author,! 
at  least,  practical  judgment  has  long  rebelled  against  the  imposi-J 
tion  of  this  measure  upon  workers  with  mental  measurements  byl 
the  experts  in  the  theory  of  measurement  of  variable  facts.  To  call  •• 
it  the  standard  deviation  has  seemed  to  him  objectionable.  There 
is  apparently  no  reason  whatever  for  its  use  except  its  supposedly 
greater  accuracy.  Perhaps  because  of  lack  of  knowledge  of  the 
purely  mathematical  side  of  statistics,  I  am  not  aware  that  this 
greater  amount  of  accuracy  has  been  calculated  from  theory  in  the 
case  of  typical  forms  of  distribution  other  than  the  so-called  'nor- 
mal.' At  all  events  it  will  be  useful  to  the  non-mathematical  stu- 
dent to  learn  the  facts  in  the  case  of  empirical  samplings  from 
known  series. 

The  series  were  A,  B,  C  and  D  of  Table  I.  The  facts  concern- 
ing the  divergences  from  the  true  average  deviation  of  the  total  series 
of  average  deviations  obtained  from  random  samplings,  and  similarly 
for  mean  square  deviations,  are  given  in  Table  II. 

The  average  deviation  and  the  mean  square  deviation  were  cal- 
culated from  an  approximate  average  never  over  a  half  of  the  unit 
of  the  scale  from  the  actual  average  and  as  a  rule  from  an  approxi- 
mate average  less  than  a  fourth  of  the  unit  of  the  scale  from  the 
actual  average.  The  Q.  was  calculated  on  the  basis  of  the  same 
suppositions  as  the  median. 

So  far  as  these  samplings  go,  the  average  deviation  is  nearly  as 
accurate  as  the  meap  square  deviation.  ^TheiaTFer~"is  on  the  whole 
5  per  cent,  more  accurate,  with  about  one  chance  in  eighteen  that  an 
infinite  number  of  drawings  from  these  series  would  raise  this  su- 
periority to  15  per  cent.  There  surely  can  not  be  enough  superiority 
of  the  latter  to  recommend  its  use  in  even  10  per  cent,  of  the< opera- 
tions involved  in  present  researches  in  psychology,  sociology  or  ^edu- 
cation. Indeed  it  is  a  question  whether  the  mathematical  statis- 
ticians ought  not  to  recognize  the  average  deviation  as  approximately 
equal  in  accuracy  and  vastly  superior  in  practical  serviceableness, 
and  hence  as  the  measure  to  be  recommended  to  students. 

There  is  something  to  be  said  in  favor  of  a  still  simpler  measure 
of  variability,  the  percentile.  Galton's  quartil^  (Q.).  for  instance 
(one  half  the  distance  between  the  25th  percentile  and  the  75th 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


percentile),  is  for  a  sampling  of  100  or  more  of  a  series  scored  on_a 
reasonably  fine  scale  nearly  as  accurate  as  the  measures  that  take 
account  of  the  amount  of  every  deviation.  The  facts  for  my  series 
are  given  in  Table  II.  In  general  the  arguments  that  support  the 
median  as  a  measure  of  central  tendency,  support  also  the  Quartile 
as  a  measure  of  variability  in  the  case  of  large  samplings  from  finely 
scaled  series  (say  of  100  or  more  cases  of  a  series  with  20  or  more 
steps).  In  the  case  of  smaller  samplings  the  calculation  of  Q.  is 
often  as  long  as  that  of  the  A.D. 

If  the  report  of  an  investigation  gives  somewhere  the  entire  dis- 
tributions the  author  may  properly  compute  only  the  medians  and 
Q.'s.  If  the  average  is  used  as  the  measure  of  central  tendency  the 
Q.  is  of  course  not  very  advantageous,  since  an  approximate  A.D. 
will  have  been  calculated  in  getting  the  average. 

Table  III.  gives  the  results  of  the  individual  sets  drawn.  It  is 
not  necessary  to  examine  it  to  follow  the  discussion  past  or  to  come, 
but  I  insert  it  for  the  sake  of  any  student  who  may  wish  to  make 
calculations  from  its  facts  other  than  those  which  I  have  made  in 
Table  II. 

TABLE    III. 
SERIES    A 


N.        Av.  M. 

101      100.2  99.8 

105     101.2  100.1 

101     100.0  99.3 

Sum  of 

Dev.          1.4  1.0 


A.D. 
10.0 
10.6 
10.4 


12.0 
13.0 
13.0 


9.3 


1.0       1.6 


N. 

101 

100 

100 

Sum  of 
Dev. 


Av. 
—  .5 
+  1.0 
+  .2 

1.7 


SEEIES    B 

M.      A.D. 
—  1.0     6.6 
+    .4     6.3 
0.0     5.9 


8.3 
7.7 
7.3 


Q. 
5.2 
5.6 
5.4 


1.4 


.8     1.1       .4 


52       99.1       98.8     11.3     14.9       8.6 
64     100.9     101.6       9.3     12.0       6.1 


54 

98.6 

99.0 

9.1 

11.2 

7.9 

51 

100.5 

99.5 

10.2 

12.6 

8.6 

Sum  of 

Dev. 

3.7 

4.3 

3.1 

4.3 

3.2 

10 

95.0 

94.0 

10.5 

11.4 

9.8 

10 

96.2 

95.0 

7.6 

9.7 

6.0 

10 

106.6 

108.0 

10.0 

12.1 

10.0 

10 

94.6 

96.0 

7.6 

9.0 

8.0 

10 

102.4 

99.0 

13.0 

17.3 

8.2 

10 

99.6 

104.0 

10.4 

11.4 

10.3 

10 

102.8 

99.0 

10.6 

11.9 

9.3 

10 

102.4 

101.0 

10.6 

12.4 

8.3 

10 

100.4 

104.0 

10.8 

12.7 

9.3 

10 

101.2 

102.0 

9.4 

14.2 

6.0 

Sum  of 

Dev. 

30.4 

36.0 

11.3 

15.9 

12.2 

50  —    .3  +    .3  6.0  7.6  5.3 

51  —2.4  —2.5  7.9  9.7  6.6 
50     +  1.7  +  1.6  5.7  7.4  4.1 
50     +    .6  +    .2  5.1  6.7  3.8 

Sum  of 


Dev. 


5.0 


4.6     3.5     2.6     4.2 


10  —  .8  +  1.0  5.0  5.7  4.3 

10  +  .6  +  .7  4.4  5.7  4.3 

10  —3.2  —2.0  6.2  8.3  4.8 

10  +1.6  +5.0  8.8  11.1  6.0 

10  —2.8  —2.0  7.0  8.3  5.5 

Sum  of 

Dev.  9.0  10.7  6.4  7.5  3.5 


MEASUREMENTS    OF    TYPE   AND    VARIABILITY 

TABLE    III.   (continued) 
SERIES   C 
Divergence  in  25  sets  of  11  cases 


Av.  Obt.-Ay. 
True.         M.  Obt.-M.  True 

—  1.0                —  1-2 
—    .8               —    .8 
—    .6               —    .5 
—    .6               —    .5 
—    .5               —    .4 
—    .4               —    -3 
—    .4               —    .3 
—    .3               —    .2 
—    .2               —    .2 
0                +    .2 
+    .1                +    .2 

In  5  sets  of 

Av.  Div. 
from  true 

55  cases 
AT.               M. 
Obt.-           Obt.- 
Tr.              Tr. 
—  .20       —    .23 
+  .09       +    .21 
+  .27       +    .32 
+  .42       +    .50 
+  .69       +    .79 

.33              .43 

A.t>. 
Obt.- 
Tr. 
.06 
.12 
.15 
.21 
.35 

.18 

Obt.- 
Tr. 
.12 
.13 
.16 
.17 
.35 

.18 

<£,.- 

Tr. 
.10 
.13 
.14 
.27 
.44 

.22 

+    -2 

+    .3 

In  3  sets  of 

110  cas< 

38 

+    .3 

+    -4 

—  .19 

—    .12 

.00 

.00 

.10 

+    .4 

+    .5 

—  .03 

—    .06 

.02 

.06 

.20 

•+    -4 

+    .5 

+  .12 

+    .33 

.09 

.06 

.33 

+    .4 

+    .5 

Av.  Div. 

+      .0 

+    .5 

from  true 

.11 

.17 

.04 

.04 

.21 

+    .5 

+    .8 

+    .6 

+    -8 

In  2  sets  of 

275  casi 

es 

+    .8 
+  1.0 
+  1.1 

+  1.2 
+  1.6 

+  1.2 
+  1.5 
+  1.5 
+  1.5 
+  1.5 

Av.  Div. 
from  true 

—  .07 

+  .02 

.045 

—    .05 
+    .06 

.055 

.04 
.06 

.05 

.07 

.08 

.075 

.04 

.07 

.055 

+  1.8 

+  3.1 

Av.  Div. 

from  true  = 

.77 


SERIES  D 

In  16  sets  of  10  cases 

AT.  Obt-AT. 

True.        M.  Obt.-M.  True. 

—  1.6  —2.5 

—  1.4  —2.0 

—  1.4  —2.0 

—  1.2  —1.0 

—  1.2  —1.0 

—  1.0  —1.0 

—  .8  —1.0 

—  .6  —1-0 

—  .4  0 

—  .4  0 

0  0 

0  0 

+    .6  +1.0 

+    .6  +1.0 

+    .8  +1.0 

+  1.8  +  2.0 
Av.  Div. 
from  true  = 

.86  1.04 


In  6  sets  of 

50  cases 

AT. 

M. 

A.D. 

<r 

Q- 

Obt- 

Obt.- 

Obt.- 

Obt.- 

Tr 

Tr. 

Tr. 

Tr. 

Tr. 

+  .09 

—    .37 

.09 

.12 

.09 

—  .76 

—    .84 

.43 

.30 

.40 

+  .16 

—    .40 

.11 

.10 

.26 

—  .10 

—    .50 

.01 

.07 

.01 

+  .36 

+    .32 

.27 

.14 

.35 

—  .56 

+  1.10 

.25 

.24 

.18 

Av.  Div. 

from  true 

.34 

.59 

.19 

.16 

.21 

In  7  sets  of 

100  casi 

BS 

+  .35 

+    .18 

.08 

.12 

.09 

—  .32 

—    .45 

.23 

.22 

.33 

+  .08 

—    .12 

.01 

.03 

.09 

+  .10 

+    .25 

.17 

.08 

.09 

+  .44 

+    .70 

.07 

.04 

.06 

—  .82 

+  1.60 

.07 

.02 

.26 

+  .22 

+    .42 

.05 

.06 

.06 

Av.  Div. 

from  true 

.33 

.53 

.097 

.081 

.140 

8  EMPIRICAL    STUDIES    OF    MEASUREMENT 

§  3.     The  Divergences  of  the  Obtained  from  the  True  Measures  by 
Theory  and  by  Experiment 

It  is  always  interesting  to  compare  the  result  of  experiments  in 
chance  with  the  expectations  derived  from  the  theory  of  probability. 
Accordingly,  I  give  the  facts  in  Table  IV.  so  as  to  save  the  reader 
interested  in  this  matter  the  time  of  collation  and  calculation  from 
the  data  of  Tables  II.  and  III. 

The  figures  under  Theory  were  calculated  not  from  the  A.D.'s 
of  all  the  separate  samplings,  but  once  for  all  from  the  A.D.  of  the 
total  series. 

The  figure  under  Theory  is  not  in  any  case  exactly  the  amount 
to  be  expected  under  strictly  correct  theory,  but  is  the  amount  to 

be  expected  from  the  formula  A.D.tr  -«bt  AT  =  —      ^5-, 

Vn 


This  formula,  applicable  to  cases  of  random  sampling  from  a 
distribution  of  the  so-called  normal  type,  will  of  course  not  suit 
exactly  distributions  limited  in  extent  and  irregular  in  form.  Com- 
parison with  it  is  however  the  important  matter  practically,  since 
it  is  the  formula  in  universal  use. 

TABLE    IV. 
Av.  DEV.  OF  OBTAINED  FROM  TRUE  AVERAGE 


ff 

A 

Theory. 

Exper. 

.C. 

¥ 

102 

3 

1.00 

.47 

47% 

100 

3 

.62 

.57 

92" 

110 

3 

.22 

.11 

50" 

100 

7 

.31 

.33 

107" 

53 

4 

1.37 

.93 

68" 

50 

4 

.87 

1.25 

144" 

55 

5 

.31 

.33 

107" 

50 

6 

.44 

.34 

77" 

10 

10 

3.20 

3.00 

94" 

10 

5 

1.9t> 

1.80 

92" 

11 

25 

.69 

.62 

90" 

10 

16 

.99 

.86 

87" 

MEASUREMENTS   OF   TYPE   AND    VARIABILITY  9 

§  4.     The  Relation  Between  the  Amount  of  a  Central  Tendency  and 

the  Amount  of  the  Variability  of  the  Group  about  the 

Central  Tendency 

In  comparing  groups  with  respect  to  variability  allowance  must 
be  made  for  the  fact  that,  in  certain  cases  at  least,  the  amounts  of 
the  central  tendency  influence  the  amounts  of  the  variabilities.  Thus 
the  A.D.  of  men  in  weight  is  hundreds  of  times  that  of  butterflies, 
yet  the  former  are  of  course  not  really  a  hundred  times  as  variable. 
Thus  the  A.D.  of  a  group  in  a  test  of  addition  was,  for  trials  of  40 
seconds,  2.18 ;  for  trials  of  80  seconds,  3.41 ;  and  for  trials  of  120 
seconds,  5.18.  It  would  obviously  be  silly  if  we  had  tested  men  with 
trials  of  80  seconds  and  women  with  trials  of  40  seconds,  and  ob- 
tained these  results,  to  infer  that  men  are  50  per  cent,  more  variable 
in  ability  to  add  than  are  women. 

In  using  the  so-called  coefficient  of  variation  (proposed  by  Pear- 
son) onemakes  allowance  for  the  possible  influence  of  the  central 
tendencies'  amounts  by  dividing  through  the  gross  variabilities  each 
by  the  amount  of  its  corresponding  central  tendency.  I  have  else- 
where shown  that  for  mental  and  social  measurements  no  one  such 
rule  can  be  always  or  even  often  right  and  suggested  that  in  any 
case  a  division  through  by  the  square  root  of  the  corresponding  cen- 
tral tendency  is  more  in  accord  with  both  theory  and  facts.1 

In  this  section  enough  data  will  be  presented  to  practically  dem- 
onstrate both  of  these  assertions.  It  is  not  important  to  investigate 
the  matter  exhaustively  for  the  very  reason  that  no  one  general  rule 
for  comparing  groups  with  respect  to  variability  can  be  found.  All 
that  is  needed  is  a  clear  enough  proof  of  the  inadequacy  of  the  prac- 
tice of  comparing  groups  after  dividing  through  the  gross  variabili- 
ties by  the  corresponding  means — clear  enough  to  stop  the  spread  of 
the  practice  and  to  warn  readers  against  conclusions  based  on  such 
comparisons. 

If  we  take  the  arrays  of  y  in  a  case  where  y  is  positively  cor- 
related with  x  we  have  a  series  of  groups  with  central  tendencies 
varying  from  lower  to  higher  which  are  selected  at  random  so  far 
as  concerns  any  influence  on  the  variability  except  the  influence  of 
the  amount  of  the  mean.  The  differences  in  variability  found  for 
these  arrays  give,  then,  in  connection  with  the  differences  in  the 
amounts  of  their  central  tendencies,  the  answer  to  our  problem  for 
the  case  of  comparisons  of  groups  with  respect  to  their  variability 
in  the  same  trait.  If  we  find  that  even  in  such  cases  there  is  no 
constant  relation  of  difference  in  central  tendency  to  difference  in 

1  Mental  gnd  Social  Measurements,  pp.  102-103. 


10 


EMPIRICAL    STUDIES    OF   MEASUREMENT 


variability,  but  that  one  law  obtains  for  stature  and  another  for 
span  or  finger  length,  then  a  fortiori  no  constant  relation  can  be  pre- 
supposed when  the  variability  of  a  group  in  one  trait  is  to  be  com- 
pared with  its  variability  (or  that  of  a  second  group)  in  a  different 
trait. 

The  first  facts  to  which  I  call  the  reader's  attention  are  the  com- 
parison of  arrays  of  y  corresponding  to  very  low  values  of  x  with 
arrays  of  y  corresponding  to  very  high  values  of  x  in  the  case  of  ten 
correlations  chosen  at  random  (so  far  as  this  issue  is  concerned) 
from  Vols.  I.  and  II.  of  Biometrika.  The  number  of  cases  ranged 
from  49  to  319.  The  results  are  given  in  Table  V.  in  the  form 
of  (1)  the  variability  of  arrays  related  to  high  central  tendencies  of 
x  (and  consequently  having  high  central  tendencies  of  y)  divided 
by  the  variability  of  arrays  related  to  low  central  tendencies  of  x, 
under  the  heading  'Gross';  (2)  the  Pearson  coefficient  of  variability 
for  the  former  divided  by  the  Pearson  coefficient  of  variability  for 

the  latter  under  the  heading  ;  and  (3)  the  similar  ratio  for  the 

0 1 

two  variabilities  each  having  been  divided  by  the  square  root  of  the 
amount  of  the  corresponding  central  tendency,  under  the  heading 

==.     A  perfect  method  would  give  values  of  100  throughout. 
VC.T. 


Width  of  head 

Length  of  left  middle  finger 

Number  of  stamens 

Number  of  stamens 


TABLE    V.   (a) 

Gross. 

101.5 

94.4 

112.3 


Frontal  breadth 

Length  of  right  antenna  (aphis) 

Number  of  stamens 


164.7 

106.8 
88.2 
111.6 


Number  of  stamens  (lesser  celandine)    132.6 


Span 

Forearm  length 

Median 


115 
95.7 
109.2 


Gross 
C.T. 

95.7 

89.5 
110 
86 

90.8 
106 
99 

90.1 


I/ C.T. 
98.5 

96.3 
135 
96.1 

100.8 

119 

107 

97.5 


Nearest  to 
Equality. 

Gross 


VC.T: 

Gross 
Gross 


Gross 
C.T. 

Gross 


Gross 
Gross 


Gross 

C.TT 

Gross 


Gross 


The  detailed  facts  from  which  these  ratios  come  are  given  in  Table  V.   (6). 


MEASUREMENTS    OF    TYPE    AND    VARIABILITY 


11 


TABLE    V.   (6) 
VABIABILITIES  OF  ARRAYS  OF  y  RELATED  TO  Low  AND  HIGH  VALUES  OF  x.    IN  TERMS  OF  A.D. 

(Each  case  measured  is  recorded  in  three  lines:  the  first  line  gives  the  values  of  x;  the 
second  line  gives  the  variabilities  of  the  related  arrays  of  y ;  the  third  line  gives  the  numbers 
of  cases  in  the  arrays.  The  volume  and  page  numbers  refer  always  to  Biometrika.) 


I.,  214 

o?  =  Head  length             18.0     18.1     18.2 

20.1     20.2     20.3     20.4 

t/  =  Head  breadth           3.2       3.1       3.0 

3.2       3.9       3.5       3.0 

35        38        51 

53        54        33        30 

I.,  216 

x  =  Height                      58.6     59.6     60.6 

69.6     70.6     71.6 

y  =  Left  middle 

finger  length              3.6       4.0       3.0 

3.2       3.3       3.2 

23        48        90 

97        46        16 

I.,  126 

#  =  No.  of  pistils           12        13        14 

20        21        22        23 

Table  I. 

y=zNo.  of  stamens         2.1       2.3       2.4 

2.8       2.6       2.3       2.4 

13        12        22 

19        13        15        10 

I.,  126 

a?  =  No.  of  pistils            678 

16        17        18 

Table  II. 

y  =  No.  of  stamens         1.8       1.1       1.1 

2.0       1.8       2.2 

6        16        35 

23        16        11 

L,  152 

x  =.  Frontal  breadth 

(aphis)  1st  brother  13.5 

19.5 

y  =  Frontal  breadth 

(aphis)  2d  brother    1.46 

1.56 

57 

50 

I.,  153 

x  =.  Length  of  antenna 

(aphis)  1st  brother  26       28 

48        50 

y  =  Length  of  antenna 

(aphis)  2d  brother    1.6       1.6 

1.2       3.3 

14        71 

43        12 

II.,  161 

x  =  No.  of  pistils 

(lesser  celandine)    13 

22 

y  =  No.  of  stamens 

(lesser  celandine)     1.8 

2.3 

24 

25 

II.,  162 

x  •=•  No.  of  pistils 

(lesser  celandine)    14        15        16 

17        28        29        30        31        32        33 

t/  =  No.  of  stamens 

(lesser  celandine)     1.9       2.6      2.1 

4.7       3.6       3.4       3.2       3.4       4.0       3.8 

10        17        16 

28        20        16        13        11        19        15 

II.,  399 

x  —  Height                      61        62 

72        73 

t/=Span                             1.2       1.4 

1.7       1.5 

8.5     32.5 

33        13 

II.,  403 

x  =  Span                         63        64 

75        76 

y  =  Forearm                        .83     1.28 

1.03     1.28 

13        32 

28        11.5 

12  EMPIRICAL    STUDIES    OF    MEASUREMENT 

The  gross  variabilities  often  increase  as  we  would  expect  with 
higher  central  tendencies,  though  by  no  means  always.  Seven  out 
of  ten  do  so,  giving  a  median  value  of  109.2  instead  of  100.  The 
Pearson  coefficient  of  variation  makes  too  much  of  a  deduction  for 
an  increase  in  the  amount  of  the  central  tendency  in  all  but  three 
cases,  giving  a  median  value  of  90.1  instead  of  100.  The  square  root 
deduction,  with  a  median  value  of  97.5,  makes  the  least  error  of  any 
one  single  method.  These  facts  alone  disqualify  the  so-called  *  coeffi- 
cient of  variation'  as  a  means  of  comparing  variabilities.  But  more 
detailed  studies  of  the  cases  of  length  of  finger,  span  and  stature  will 
be  still  clearer. 

The  facts  for  length  of  left  middle  finger  are  as  given  in  Table  VI. 

TABLE    VI. 

RELATION  OF  AMOUNT  OF  VARIABILITY  TO  AMOUNT  OF  CENTRAL  TENDENCY. 
FINGEE  LENGTH.     (Biometrika,  Vol.  I.,  p.  216) 


Array. 
1 
2 
3 
4 
5 


9 
10 
11 
12 
13 
14 
15 
16 
17 

In  the  case  of  finger  length  increase  in  the  amount  of  the  central 
tendency  does  not  imply  an  appreciable  increase  in  the  amount  of 
variability.  No  allowance  is  needed. 

In  the  case  of  span  it  would  be  equally  absurd  not  to  make  an 
allowance  and  one  as  great  or  nearly  as  great  as  the  Pearson  method 
makes.  For  the  preliminary  study  of  the  variability  of  span  re- 
ported in  Table  V.  is  confirmed  by  the  facts  in  the  case  of  three 
other  span  series.  These  facts  (given  in  Table  VII.)  abundantly 
prove  that  the  influence  of  the  amount  of  the  central  tendency  on  the 
amount  of  the  variability  follows  totally  different  laws  in  the  case 
of  span  ana  of  ringer  length. 


Value  of  x  to  Which 
the  Array  is 
Related. 
581 

No.  of  Cases  in 
the  Array. 
6 

Central  Ten- 
dency of  the 
Array. 
103 

Variability  (A.D.) 
of  the  Array. 

167 

591 

23 

107 

357 

601 

48 

108 

404 

611 

90 

109 

309 

621 

175 

111 

325 

631 

317 

112 

347 

641 

393 

114 

312 

651 
661 

920 

116 

331 

671 

413 

118 

339 

681 

264 

119 

345 

691 

177 

120 

334 

701 

97 

122 

322 

711 

46 

124 

333 

721 

17 

126 

318 

731 

7 

128 

386 

741 

4 

128 

275 

MEASUREMENTS    OF    TYPE    AND    VARIABILITY 


13 


TABLE    VII. 

RELATION  OF  AMOUNT  OF  VARIABILITY  TO  AMOUNT  OF  CENTRAL  TENDENCY. 
SPAN.     (Biometrika,  Vol.  II.,  pp.  399-401) 


Daughters. 
N.         C.T.     Var. 


Fathen 

i. 

SODS. 

Mothers. 

N. 

C.T. 

Var. 

N. 

C.T. 

Var. 

N. 

C.T. 

Var. 

32.5 

642 

578 

31 

657 

519 

18 

571 

428 

42.5 

651 

587 

56 

660 

505 

34.5 

582 

596 

71.5 

658 

560 

78.5 

670 

516 

79.5 

593 

600 

122.5 

667 

666 

127 

677 

580 

135.5 

600 

524 

142.5 

675 

662 

178.5 

687 

608 

163 

609 

608 

136.5 

687 

593 

189 

700 

600 

183 

619 

573 

154.5 

692 

574 

137 

707 

636 

163 

627 

554 

118.5 

702 

658 

137 

715 

505 

114.5 

637 

542 

102.5 

713 

698 

93 

720 

601 

78.5 

640 

624 

56.5 

720 

601 

52.5 

735 

503 

41 

647 

588 

33 

735 

678 

39 

745 

595 

16 

655 

881 

15.5 
52 

101 

150 

199 

438 

169.5 

151.5 
81.5 
40.5 
19.5 


585  471 

595  447 

600  466 

613  515 

620  485 

625  510 

650  492 

660  605 

660  601 

665  481 

680  436 


As  a  final  case  let  us  take  stature.  Here  the  variability  is 
slightly  less  as  the  amount  of  the  central  tendency  increases.  The 
facts  are  given  in  Table  VIII.  constructed  on  the  same  plan  as 
Table  VI. 

TABLE    VIII. 

RELATION  OF  AMOUNT  OF  VARIABILITY  TO  AMOUNT  OF  CENTRAL  TENDENCY  IN 

GROUPS  DIFFERING  IN  CENTRAL  TENDENCY.     STATURE. 

(Biometrika,  Vol.  I.,  p.  216) 


Related  to  x. 
10.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 
11.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 
12.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 
13.0 


44 

74 

177 
315 
347 
461 
458 
346 
289 

180 

44 
52 
35 
31 
25 
-7 
8 


C.T. 
61.1 
61.1 
60.6 
60.7 
62.3 
62.8 
62.8 
62.1 

63.6 


64.6 
65.1 
66.1 
66.1 
67.1 

67.1 

67.6 
69.6 
69.6 
68.6 
70.1 
69.1 
69.1 


A.D. 
286 
146 
190 
179 
170 
183 
132 
153 

137 


155 
152 
158 
156 
147 

157 

170 
158 
147 
127 
148 
136 
264 


14  EMPIRICAL    STUDIES    OF    MEASUREMENT 

MEASUREMENTS  OF  EELATIONSHIPS 

The  importance  to  any  science  of  exact  and  convenient  methods 
of  measuring  the  relationships  of  the  facts  it  studies  should  be 
obvious.  It  is  therefore  unfortunate  that  students  of  psychology 
and  the  social  sciences  have  with  few  exceptions  neglected  both  the 
theoretical  problem  of  correlated  variations  and  the  careful  measure- 
ment of  such  relationships  as  they  have  in  fact  found. 

The  failure  to  utilize  the  methods  devised  by  Galton,  Pearson, 
Sheppard,  Spearman  and  others  is  due  partly  to  an  ignorant  and 
partly  to  an  intelligent  suspicion  aroused  by  the  mathematical 
derivations  of  these  methods.  Ignorance  of  the  rationale  of  their 
derivations  cooperating  with  ignorance  of  the  conditions  which  re- 
quire their  use  and  of  the  necessity  of  some  such  refined  methods  has 
caused  the  stupid  suspicion  and  aversion.  Inability  to  follow  the 
mathematics  of  the  derivation  of  formula?,  at  least  in  detail,  cooperat- 
ing with  the  rational  expectation  that  too  abstract  methods  will  fit 
the  concrete  cases  imperfectly  and  with  the  equally  rational  con- 
fidence that  proofs  resting  upon  the  assumption  of  close  approxima- 
tion of  actual  variations  in  mental  and  social  facts  to  the  probability 
curve  distribution  are  always  unsafe  and,  perhaps,  usually  mislead- 
ing, has  caused  the  intelligent  suspicion. 

It  is  probable  that  unless  these  methods  are  soon  subjected  to  a 
review  by  some  one  who  can  both  make  perfectly  clear  their  presup- 
positions to  the  rank  and  file  of  investigators  in  psychology  and  the 
social  sciences  and  prove  their  applicability  to  actual  cases  of  rela- 
tions to  be  measured,  there  will  be  damage  done  in  two  ways.  Many 
investigators  will  as  in  the  past  use  hopelessly  crude  methods  and 
misinterpret  relationships;  and  also  many  investigators  will  learn 
off  the  formulas  of  the  mathematical  statisticians  and  apply  them 
to  cases  where  they  are  out  of  place  and  give  inadequate  and  mis- 
leading results.  To  both  of  these  errors  the  writer,  for  instance, 
confesses  himself  guilty  in  the  past. 

I  am  unable  to  make  such  a  review  but  as  no  one  of  those  who 
are  able  seems  willing,1  I  have  made  a  partial  and  inferior  substi- 
tute for  it  which  I  hope  may,  in  so  far  as  it  is  sound,  be  instructive 
to  students  of  mental  measurements  and,  in  so  far  as  it  is  unsound, 

1  Perhaps  Mr.  C.  Spearman's  article  on  '  The  Proof  and  Measurement  of 
Association  between  Two  Things'  (in  the^jw.  J.  of  Psy.,  Vol.  XV.)  may  be 
considered  as  filling  the  need,  but  I  fear  that  it  is  too  technical  in  parts  and 
not  inquisitive  enough  concerning  the  actual  relations  between  (1)  the  indi- 
vidual relationships,  from  which  all  our  computations  ought  to  start,  and  (2) 
the  general  expressions  or  summaries  of  them.  At  all  events  I  am  not  trying 
to  do  over  again,  for  better  or  worse,  what  Mr.  Spearman  has  done,  but  some- 
thing which  is  needed  as  introductory  and  accessory  to  his  work. 


MEASUREMENTS    OF    RELATIONSHIPS 


15 


may  provoke  some  capable  student  to  give  the  adequate  review  that 
is  so  much  needed. 

This  report  will  presuppose  in  the  reader  knowledge  of  the  bare 
elements  of  the  theory  of  measurement  of  variable  facts  such  as  is 
given  for  instance  in  the  writer's  Introduction  to  the  Theory  of  Men- 
tal and  Social  Measurements.  It  will  deal  in  order  with  the  fol- 
lowing topics : 

I.  What  is  actually  measured  by  typical  measures  of  the  relation- 
ship between  first  and  second  member  of  a  pair  in  a  series  of  pairs 
of  values,  each  first-member  value  being  a  deviation  from  the  central 
tendency  of  one  series  and  each  second-member  value  being  a  related 
deviation  from  the  central  tendency  of  a  second  series? 

II.  What  are  the  respective  presuppositions  of  each  of  these 
typical  measures? 

III.  What  are  the  advantages  and  disadvantages  of  each  of  these 
typical  measures? 

The  only  original  contributions  which  this  discussion  contains 
are  (1)  the  investigation  of  certain  artificially  constructed  cases  of 
correlation  and  (2)  a  laborious  but  not  very  important  experimental 
testing  of  the  comparative  reliability  of  different  measures  of  rela- 
tionship, and  (3)  a  similar  experimental  testing  of  methods  for  cor- 
recting measures  of  relationship  for  the  'attenuation'  due  to  inaccu- 
rate original  data. 


§  5.  I.  What  is  actually  measured  by  typical  measures  of  the 
relationship  between  first  and  second  member  of  a  pair  in  a  series 
of  pairs  of  values,  each  first-member  value  being  a  deviation  from 
the  central  tendency  of  one  series  and  each  second-member  value 
related  deviation  from  the  central  tendency  of  a  second  series 

Consider  the  following  series  of  paired  values  of  A  and  B : 


A 

—  1 

—  5 
3 


—  5 

—  5 

—  3 

—  1 

—  7 


3 

o 

—  3 

—  1 


A 

_  I 
_  J 
_  1 

—  1 


+  1 

+  1 


—  3 

—  1 
+  1 
+  1 
+  3 

—  3 

—  1 
+  1 
+  3 
+  5 


B 

+  7 


+  3         - 


+  3 
+  3 
+  3 
+  5 
+  5 
+  5 
+  7 


+  1 
+  3 
+  5 
—  1 
+  3 
+  3 
+  5 


—  3 

—  1  +1 

'.+•  1  +  1 

—  5  +1 

Pearson  Coefficient  =.634. 
Median  Ratio  B/A  =  .65. 
Average  of  Ratios   =.902. 

The  average  of  ratios  is  valueless  because  it  overweights  positive  values  of 
2  pairs,  etc.  A 

Per  cent,  unlike  signs  =  .267,  r  as  calculated  therefrom  being  .665.  [Mi,  ff" 


16  EMPIRICAL    STUDIES    OF    MEASUREMENT 

Each  of  these  pairs  represents  a  relationship,  the  entire  series 
reading:  A  deviation  in  A  of  —  7  from  the  central  tendency  of  A 
brought  with  it  a  deviation  in  B  of  —  5  from  the  central  tendency  of 
B;  a  deviation  of  —  5  brought  in  one  case  a  deviation  in  B  of  —  5, 
in  a  second  case  one  of  —  3,  and  in  a  third  case  of  —  1,  etc. 

Consider  now  two  measures  each  expressing  an  important  fact 
concerning  this  series  of  30  individual  relationships.  The  first  is, 

.634.      The  second  is,   The  median  of  the  30  B/  A 


ratios  =  .65.  The  former  is  of  course  the  Pearson  Coefficient  of 
correlation  for  A  —  B;  the  latter  is.  the  Median  or  Mid  Ratio  B/A. 

What  the  former  measures  can  not  be  stated  except  in  terms  not 
yet  given  by  the  individual  relationships  themselves.  Professor 
Pearson's  own  statements  for  instance  are  in  terms  of  certain  facts 
of  a  correlation  diagram  such  as  Fig.  1,  not  in  terms  of  the  indi- 
vidual relationships. 

It  is  clear  that  in  the  case  of  Fig.  1,  which  represents  our  30 
relationships  graphically,  the  slope  of  the  straight  line  LL1  through 


-7  -S  -3  -I 


-3 


+3 


+7 


O  so  drawn  that  the  sum  of  the  deviations  of  the  individual  dots  from 
it  is  zero  (measuring  deviations  in  the  direction  of  the  B  line  and 
calling  deviations  above  the  line  in  the  left  hand  half  of  the  surface 
and  below  the  line  in  the  right  hand  half  of  the  surface  +,  and 
calling  deviations  below  the  line  in  the  left  half  and  above  the  line 
in  the  right  hand  half  —  )  is  a  measure  of  an  important  fact  about 
the  series  of  relationships. 

I  The  Pearson  Coefficient  does  not,  however,  measure  the  slope  of. 
/  just  such  a  line  as  we  have  supposed  to  be  drawn  in  Fig.  1  and 
I  described  in  the  last  paragraph.  Its  line  is  not  so  calculated  as  to 

1  In  this  case  the  slope  is  roughly  73  per  cent,  of  45°,  the  slope  which  would 
be  found  were  correlation  perfect.  The  slope  for  the  A's  taken  as  dependent  on 
the  B's  is  roughly  64  per  cent,  of  45°. 


MEASUREMENTS    OF   RELATIONSHIPS 


17 


make  the  deviations  from  it  toward  closer  correlation  equal  to  the  ' 
deviations  from  it  towards  less  correlation,  but  is  so  calculated  as 
to  make  the  sumof  the  squares  of  the  deviationsT-from  it  least 

This  of  course  weights  the  extreme  deviations  much  more  than 
those  near  the  jenterof  the  ..sn^fapp^  f°r  the  same  change  in  the 
slope^oFthe Ime  alters  the  sum  of  the  squares  of  the  deviations  from 
the  line  near  the  center  of  the  surface  far  less  than  that  of  the  re- 
mote deviations.  This  is  a  possibly  questionable  feature  of  the 
Pearson  Coefficient. 

Moreover  it  is  calculated  as  the  slope  of  this  line  of  so-called 
'  regression '  as  found  when  the  two  traits  are  reduced  to  equivalence 
of  variability  and  double  entries  are  made  in  the  correlation  table, 
*.  e.,  B's  as  related  to  A's  and  A's  as  related  to  B's,  the  two  sets 
of  entries  being  so  superposed  that  the  intersection  of  the  means  in 
the  one  case  coincides  with  the  intersection  of  the  means  in  the 
other  case. 

Professor  Pearson  gives  many  readers  the  impression  that  his 
coefficient  of  correlation  is  calculated  as  the  slope  of  the  straight  line 


Fi 


F(3.  3. 
-7   -5"   -3    -I    +1   +3   +S   +1 


-S 
-3 
-I 
+  l 
+3 


through  0  to  fit  the  points  in  the  correlation  diagram  that  represent 
the  means  of  the  arrays1  (the  two  related  series  being  reduced  to 
an  equivalence  in  variability  and  entered  doubly),  but  in  fact  it 
is  the  slope  of  the  line  from  which  the  sum  of  the  squares  of  the 
deviations  of  all  the  dots  each  representing  one  relationship  is  least, 
not  the  slope  of  the  line  from  which  the  sum  of  the  squares  of  the 
deviations  of  the  dots  representing  each  the  mean  of  one  array  is 
least.  It  is  in  onr  illustration  a  line  to  fit  the  dots  of_Fig.  3rjnot 
fhnsy  ftf  Figr.  2.  That  is,  an  array  of  100  cases  is  (quite  properly) 


given  greater  weight  than  one  of  2  cases. 

1  See,  for  instance,  '  Grammar  of  Science,'  2d  edition,  1900,  p.  393  and  p.  396. 


18 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


Consider  now  the  Pearson  Coefficient  from  another  point  of  view. 
Let  us  for  the  present  restrict  relationships  to  those  between  two 
series  of  the  same  form  of  distribution,  and  also  define  perfect  corre- 
lation  as  a  relationship  such  that  any  deviation  of  A  from  its  central 
tendency  will  imply  a  deviation  of  B  from  B's  central  tendency 
which  shall  be  the  same  fraction  of  B  7s  variability  that  the  deviation" 
of  A  is  of  A 's  variability!  That  is, 


A, 


3-z,     etc. 


Var.  of  B  series      Var.  of  A  series'    Var.  of  B      Var.  of  A' 

j-  -j  j  t.    Var.  of  5  series  .      ,,  . 

If  then  all  values  of  B  are  divided  by  TT.    —  „       — .  -  ,  we  should  in 

Var.  of  A  series 

perfect  correlation  find  each  deviation  of  A  accompanied  by  an 
identical  deviation  of  B.  The  sum  of  the  AB  products  would  be 
equal  to  the  sum  of  the  A2,  or  to  the  sum  of  the  B2,  or  to  V2A2  V5B2. 

In  the  case  of  two  series  of  the  same  form  of  distribution  and 
of  equal  variability  the  Pearson  Coefficient  formula  then  measures 
the  proportion  which  the  sum  of  the  series  A^B^  A2B2,  etc.,  is  of 
what  it  would  be  with  perfect  correlation  as  defined. 

It  can  be  shown  that  without  reducing  B  or  A  to  equivalence  in 
variability  perfect  correlation  as  defined  would  give  for  the  sum 
of  the  AB  products  V2A2  V2.B2,  provided  the  form  of  distribution 
of  A  is  the  same  as  that  of  B. 

The  Pearson  Coefficient  measures,  then,  in  cases  where  the  form 
of  distribution  of  the  two  facts  to  be  related  is  the  same,  the  propor- 


tion which  foe  sum  of  the  AB  products  is  of  what  it  wouldJae_were 


correlation  ^perfect. 


There  is  no  ambiguity  as  to  what  is  measured  by  the  median  of 
the  B/A  ratios.  Whatever  the  distributions  may  be  or  the  ratios, 
the  median  means  always  a  definite  thing:  the  ratio  B/A  which  is 
exceeded  in  magnitude  by  as  many  of  the  ratios  as  it  exceeds.  We 
have  only  to  note  that  the  median  of  the  B/A  's  and  the  median  of 
the  A/B's  are  two  different  things  and  that  if  we  are  interested  in 
representing  in  one  number  both  what  a  given  A  deviation  implies 
with  respect  to  B  and  what  a  given  B  deviation  implies  with  respect 
to  A,  we  must  use  both  the  B/A  and  the  A/B  median. 

Certain  other  measures  deserve  mention.  The  directly  calcu- 
lated average  of  all  the  individual  relationships  B/A  or  A/B  is  a 
perfectly  comprehensible  measure  but  rather  a  useless  one.  The 
Modal  Ratio  B/A  or  A/B  is  also  a  perfectly  clear  conception  and, 
in  cases  where  it  can  be  easily  and  accurately  determined,  a  very 
valuable  one. 

The  per  cent,  of  direct  or  the  per  cent,  of  inverse  relationships 
i$  equally  comprehensibly  ami  is  an  important,  fnnctinn  nf  tt 
ness  of  relationship. 


MEASUREMENTS    OF    RELATIONSHIPS 


19 


—39  etc. 


TABLE    IX. 

—1  +1 


+39 


39 

1 

37 

1 

35 

1 

1                                              1 

1                                    1        1 

1 

1                           111 

1111                 2 

1             1 

11                           1111             1 

, 

1      1 

1        11             11111                 1         1 

1  1 

1        1         121        111         111        1 

111            1 

111        1        122221                  1        1 

1         21 

2        3112221111        1        2                  1        1 

11      1      1 

221122122112221         1        1                 1 

2  1  1 

133233323312        1        3 

12      2 

52224422331312        21 

1 

1243535334441111        11        1 

1  1 

11425536434342121        2 

5 

1  1      1 

211345463643        322        22        11        1 

3 

1 

1224546646473211        1        1        1 

1 

1 

1 

1121325346584642221 

1        111325155444543331111      21 

5 

1 

1        135433444655231111      1 

11        2133334464542221               3 

1             1             1        33244263533512       1 

212233232234342211             1 

1 

1        1             2222223451322       1      1 

11        1        2111223234        3211 

1 

1                  1             111121212        121        211111 

1             1        22211121        1        2111 

1                  1                  1112211                  21                     1 

11        1            1111111        1        1 

11111             1             1                  11 

1                          1                  1111 

1111                  1 

1        1        1 

1                1 

1 

1 

39 

1 

111  2356912  1620263137435054596263  63625954504337312620151296  532111 

When  the  individual  values  of  A  and  B  are  not  measured  as 
amounts  of  deviation  from  their  central  tendencies,  but  only  as  so 
many  AlJs  known  to  be  less  than  Z  and  so  many  A2's  greater  than 
Z,  and  as  so  many  B^'s  less  than  W  and  so  many  J32's  greater  than 
W,  the  per  cents,  of  A1^1  pairs,  A*B2  pairs,  A2^1  pairs  and  A2B2 
pairs  give  important  information. 

The  number  and  amount  of  the  divergences  of  the  ranks  of  the 
second  members  from  the  ranks  of  their  related  first  members  also 
give  important  information. 

If  the  two  related  facts  are  of  the  so-called  normal  distributionl 
and  the  relationship  is  uniform  for  all  amounts  of  A  and  each  array! 
is  also  a  normal  distribution,  the  Median  Ratio,  the  Modal  Ratio  and! 


20 


EMPIRICAL    STUDIES    OF   MEASUREMENT 


X 


\ 


FIG.  4. 

the  Pearson  Coefficient  will,  if  the  two  series  are  reduced  to  equiv- 
alence in  variability,  coincide  and  will  equal  cosine  wf/.1  This  is 
the  case  of  so-called  normaTcUTfeTation  approximated  in  many  or- 
ganic and  hereditary  anatomical  relationships.  It  is  of  course  only 
one  of  many  possible  types  of  relationship.  The  extent  to  which  it 
prevails  in  mental  and  social  relationships  is  not  known.  Its  pre- 
valence in  the  case  of  anatomical  facts  has  probably  been  over- 
estimated. 

Table  IX.  gives  the  facts  of  the  relationship  between  two  series 
both  of  the  same  form  of  distribution,  almost  exactly  the  so-called 
normal,  and  of  the  same  variability,  the  relationship  being  devised 
artificially  so  that  the  average  of  each  array  of  y  is  .5  X  the  corre- 
sponding value  of  x.  This  regression  of  y  on  x  is  shown  graph- 
ically in  Fig.  4,  which  gives  the  average  of  each  array  of  the  i/'s. 
The  regression  of  x  on  y  is  shown  graphically  in  Fig.  5,  which  gives 
the  average  of  each  array  of  the  x's.  The  Pearson  Coefficient  for 
this  case  is  .53.  The  Median  Ratio  is  much  higher  (.60  for  the  y/x 
1  U  equalling  the  per  cent,  of  unlike-signed  pairs. 


MEASUREMENTS   OF   RELATIONSHIPS 


21 


and  x/y  ratios  together)  because  the  correlation  is  much  closer  for 
mediocre  values  of  x  and  y  than  for  extreme  values  (see  especially 
the  regression  of  x  on  y}.  U  is  .292  and  r  from  cos  «T7  is  accorcU. 
ingly  .61. 

This  case  illustrates  the  fact  that  the  relation  of  y  to  x  may  not 


be  the  same  as  that  of  x  to  if  even  when  the  form  of  distribution  and 
variability  is  the  same  for  both  cases.  It  also  illustrates  a  rather 
close  approach  to  the  so-called  'normal'  correlation. 


FIG.  5. 

Table  X.  gives  graphically  the  correlation  in  the  case  of  age  at 
death  of  husband  with  age  at  death  of  wife  in  935  pairs  from 
records  of  the  Society  of  Friends.  This  is  taken  from  the  table 
on  p.  498  of  Vol.  I.  of  Biometrika,  the  table  being  due  to  Mary 
Beeton  in  cooperation  with  Karl  Pearson.  This  case  shows  a  rela- 
tionship between  two  series  neither  of  which  is  anything  like  normal 
in  form  of  distribution,  which  are  not  of  the  same  form  of  distribu- 
tion and  which  therefore  are  in  strictness  incomparable  in  varia- 
bility. 


22 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


Age  of  Husbnd. 
11-11  M     M-3/eK. 


I    I 


'  I  I 


I 


'  I 

I  I 

I  i 

I  I 

I  I 


I        I 


TABLE    X. 

Fig.  6  gives  the  regression  of  y  (wife's  age)  on  x  (husband's 
age)  in  terms  of  averages  of  arrays  of  y  and  also  of  medians  of 
arrays  of  y.  To  give  the  regression  by  single  modes  for  the  arrays 
would  be  fallacious,  for  each  array  is  more  or  less  clearly  a  bimodal 
distribution.  This  is  shown  in  Fig.  7,  where  the  s/'s  are  grouped 
in  four  large  arrays.  It  should  be  clear  that  any  single  figure  is 
inadequate  to  express  this  relationship.  The  Pearson  Coefficient  of 
correlation  is  .20  and  the  regression  of  y  (wife's  age)  on  x  (hus- 
band's age)  calculated  from  it  is  .25.  But  this  would  lead  one  far 
astray  concerning  the  real  regression,  as  we  see  by  Fig.  6.  The 
relationship  is  closer  for  early  deaths  than  for  late.  The  form  of 
distribution  of  the  relationship  is,  apart  from  this,  skewed  in  gen- 
eral from  a  mode  of  close  resemblance  toward  very  great  diversity, 
and  is  in  the  third  place  complicated  by  the  submodal  tendency  of 
a  wife  to  die  at  about  35  more  often  than  at  30  or  40.  Jguch  a  case 
illustrates  the  fact  t.ha.f.  panTi  typp  nf  measure  of  a  relationship  meas- 


ures some  particular  aspect  thereof  and  also  the  fact  of  the  extreme 
{jbstractness  from  realityjjf  the  Pearson  Coefficient,  which  in  this 


MEASUREMENTS   OF   RELATIONSHIPS  23 

case  measures  neither  a  uniform  tendency  nor  a  central  tendency 
of  the  series  of  individual  relationships. 

The  reader  will  obtain  concrete  information  about  the  meaning 
of  the  different  measures  of  relationship  and  of  their  merits  in  actual 
practise  if  he  will  calculate  them  for  a  score  of  representative  rela- 
tionships and  examine  them  in  the  light  of  the  entire  correlation 
tables.  I  have  done  this  for  the  cosine  irU  and  Median  Ratio  (or  rather, 


X   Age   o(   Musi/and 


FIG.  6.     The  dotted  line  is  from  averages ;  the  continuous  line  from  medians. 
The  dash  line  is  the  regression  as  calculated  from  the  Pearson  Coefficient. 

in  order  to  have  the  resulting  figure  comparable  directly  with  the 

At     •y'QT*        'P 

cosine  irU  and  the  Pearson  r,  for  the  median  of  all  the  ratios : — 

and  x  var'  y  \  in  the  case  of  nine  relationships  representing  organic 
y  var.  x) 

and  hereditary  and  conjugal  relations,  relations  in  animals  and  in 
plants,  relations  of  definite  structural  features  and  complex  prop- 
erties. The  results  are  given  in  Table  XI.  They  show  that  the 


24 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


median  ratio  method  gives  results  as  close  to  the  unlike-signs  method 
as  does  the  Pearson  method.  The  reader  who  will  examine  Table 
XL  in  connection  with  the  original1  correlation  tables  in  Biometrika 

il-43     *K-S8       SI- If  71-103 


Vol. 
I., 
I., 

L, 

I., 

II., 
II., 
II., 
II., 
III., 


FIG.  7. 

will  find  also  that  where  the  Pearson  Coefficient  r  and  the  Median 
Ratio  r  diverge  at  all  widely  it  is  the  latter  which  better  fulfils 
Pearson's  criterion  of  telling  how  much  nearer  the  most  probable 
value  of  a  second  member  of  a  pair  is  to  the  value  of  the  first  mem- 
ber than  it  would  be  with  no  relationship  at  all. 

TABLE  XI. 


Page  Traits  to  be  Related 

84     Longevity  of  adult  brothers 
126     No.  of  stamens  with  No.  of  pistils  in 
late  flowers  of  Ficaria  ranunculoides 
214     Human  head  length  with  head  width 
216     Human    height    with     left    middle 

finger  length 
97     Capsule    height    of   brother    plants 

(Shirley  poppies) 
97     Stigmata  of  brother  plants  ( Shirley 


163     NoVof  stamens  with  No.  of  pistils  in 

lesser  celandine  from  Surrey 
498     Longevity  of  husband  with  longev- 
ity of  wife.     Friends'  records 
170     Cephalic  index  of  brothers 
Average  difference  of  r  by  Pearson  Coefficient  from  r  by  cos  wU  .055. 
Average  difference  of  r  by  Median  Ratio  from  r  by  cos  irU  .045. 

1  These  examples  are  all  taken  from  the  first  three  volumes  of  Biometrika, 
the  '  Vol.'  and  '  Page '  of  the  table  referring  to  that  journal. 


x 

y 

Mutual  Relationship 
By           By          By 
Pearson    Median  Cosine 
N       Coef.        Ratio       nU 
2000     .2853     .479     .3763 

Pistils 
Length 

Stamens 
Width 

373 
3000 

.7489 
.4016 

.80 
.415 

.7815 
.3875 

Height 

L.M.F. 

3000 

.6608 

.69 

.6747 

13800 

.3782 

.48 

.5030 

4716 

.2561 

.253 

.2160 

Pistils 

Stamens 

500 

.6601 

.55 

.5570 

Husband 

Wife 

935 
1982 

.1999 
.49 

.41 
.53 

.2560 
.5090 

MEASUREMENTS    OF   RELATIONSHIPS  25 

§  6.     The  Presuppositions  of  Measures  of  Relationship 
The  Pearson  Coefficient. 

Taken  at  its  mere  face  value,  — —     -  or  ,  the  Pearson 

V2x2  VSt/2        Tioi^ 

Coefficient  has  of  course  no  presuppositions,  but  if  it  means  the 
proportion  that  the  2(xy)  is  of  what  it  would  be  with  perfect  corre- 
lation it  presupposes  sameness  of  form  of  distribution  in  the  two 
series.  If  it  means  the  proportion  which  the  slope  of  a  certain 
straight  line  is  of  the  slope  of  the  line  of  perfect  correlation,  the 
certain  line  being  so  drawn  that  the  sum  of  the  squares  of  the 
divergences  from  it  of  the  given  y  values  (in  double  entry)  toward 
greater  correlation  equals  the  sum  of  the  squares  of  those  toward 
less,  it  presupposes  the  'normal'  distribution  in  the  case  of  both 
series. 

The  Median  Ratio. 

The  Median  Ratio  need  have  no  presuppositions.  It  is  simply 
one  of  the  obtained  individual  relationships.  When,  however,  we 
come  to  draw  inferences  from  it  about  the  entire  series  of  relation- 
ships, we  must  state  certain  additional  facts  or  use  certain  presup- 
positions. 

The  Modal  Ratio  and  the  Percentage  of  Like-signed  or  of  Un- 
like-signed  pairs  are  also  directly  drawn  from  the  series  of  indi- 
vidual relationships  themselves.  In  calculating  the  general  trend 
of  relationship,  r,  from  r=  cosine  irV  (U  being  the  per  cent,  of  un- 
like-signed  pairs)  we  presuppose  (if  I  understand  Mr.  Sheppard  cor- 
rectly) that  the  correlation  surface  is  transformable  into  a  surface 
of  revolution  by  a  slide  and  two  stretches. 

§  7.     The  Advantages  of  the  Different  Measures 
The  two  previous  sections  are  preliminary  to  the  main  topic 
which  forms  the  title  of  this  section. 

I  shall  first  compare  the  conventional  measure,  the  Pearson  Coeffi- 
cient, with  the  Median  Ratio  and  later  deal  very  briefly  with  some  of 
the  other  measures. 

The  main  desiderata  in  any  measure  are  that  it  measure  some 
real  fact  and  that  this  fact  be  important!  Other  desiderata  in  the 
case  of  a  measure  of  relationship  are  that  the  measure  be  comparable 
with  other  measures  of  other  relationships,  that  it  be  conveniently 
and  easily  calculated  and  that  it  diverge  little  from  the  correspond- 
ing measure  of  the  total  series  from  a  random  sampling  of  which 
it  is  calculated.  These  desiderata  we  will  consider  in  the  above  order. 

Reality. 

The  Median  Ratio  is  a  clear  statement  of  a  real  fact,  an  observed 


26  EMPIRICAL    STUDIES    OF    MEASUREMENT 

relationship,  suchthatthe  number  of  relationships  closer  than  it 
e(fuals  the  number  less  close.     It  gives  the  amount  of  i/'s  difference 


from  its  central  tendency  implied  by  such  difference  in  x  for  this 
mid-case. 

The  Pearson  r  is  not  an  observed  relationship  but  a  measure  in- 
ferred from  certain  features  of  the  observed  relationships  on  the 
basis  of  certain  presuppositions  about  them  and  the  distribution 
of  the  facts  from  which  they  come.  It  is  of  course  real  in  the  sense 
of  being  the  most  probable  real  central  tendency  of  the  relationships 
if  these  various  presuppositions  are  true,  but  in  fact  they  never  are 
except  by  chance  more  than  approximately  true,  and  in  the  majority 
of  the  cases  in  which  students  of  the  mental  and  social  sciences  need 
to  measure  relationships,  they  are  far  from  true. 

The  'regression,'  that  is  the  relation  between  actual  amounts  of 
y  and  actual  amounts  of  x,  is  the  reality  at  the  basis  of  all  measures 
of  the  relationship.  The  Median  Ratio  expresses  it  directly.  It  can 
be  ascertained  from  the  Pearson  r  only  indirectly  and  on  the  hypoth- 
esis that  certain  very  questionable  conditions  are  realized. 

Importance  of  the  Fact  Measured. 

There  is  no  great  advantage  either  way  in  this  respect.  Neither 
the  Pearson  Coefficient  nor  the  Median  Ratio  gives  the  entire  fact 
of  the  relationship.  Only  the  total  distribution  of  the  relationship 
that.  For  'normal'  correlation  where  the  relationship  is  the 


same  regardless  of  the  amount  of  x  and  where  all  of  the  arrays 
are  distributed  in  normal  surfaces  of  frequency  the  Pearson  Coeffi- 
cient and  the  Median  Ratio  both  give  the  central  tendency  of  the  rela- 
tionship. In  other  cases  than  this  the  Median  Ratio  is  a  trifle  more 
important  because  less  misleading  and  because  it  is  nearer  the  modal 
relationship  if  the  distribution  of  the  relationship  is  skewed. 

It  is  also  worthy  of  note  that  our  thinking  about  relationships 
should  for  practical  reasons  usually  be  in  terms  of  the  actual  y/x 
or  x/y  ratios,  that  is  the  'regressions,'  since  what  we  usually  need 
to  know  is  the  implication  of  some  actual  deviation  of  one  concern- 
ing the  related  deviation  of  the  other.  It  seems  better  then  to 
calculate  the  y/x  or  x/y  ratio  directly  and  when  necessary  to  infer 
the  r  (that  is  the  ratio  when  both  traits  are  reduced  to  an  equivalence 
in  variability  and  the  correlation  table  is  one  of  double  entry)  rather 
than  to  calculate  the  r  and  infer  the  y/x  or  x/y  ratio. 

Comparability. 

To  compare  the  relationship  between  A  and  B  with  that  between 
C  and  D  adequately,  we  must  compare  the  total  distribution  of  the 
relationship  A  —  B  with  the  total  distribution  of  the  relationship 
C  —  D.  The  Pearson  Coefficients  of  A  —  B  and  C  —  D  are  per- 


MEASUREMENTS    OF   RELATIONSHIPS  27 

fectly  fit  to  compare  only  when  the  form  of  distribution  of  the 
relationship  A  —  B  is  the  same  as  that  of  the  relationship  C  —  D. 
So  also  of  the  median  B/A  and  median  D/C,  or  of  the  median 
A/B  and  median  C/D,  or  of  any  measure  of  the  central  tendency 
of  relationship  which  may  be  inferred  from  them.  In  so  far  as  what 
we  wish  to  compare  is  the  modal  relationship,  however,  there  is  a 
smaller  error  as  a  rule  in  inferring  from  the  comparison  of  the 


Median  Ratios  _of  unlike  distributions  of  relationships  than  in  in- 
ferring from  the  comparison  of  their  Pearson  Coefficients. 

Convenience  of  Calculation. 

Provided  the  original  measures  are  on  a  sufficiently  fine  scale, 
as  they  ought  for  every  reason  to  be  where  relationships  are  to 
be  measured  by  a  Pearson  Coefficient  or  a  Median  Ratio  or  a  Modal 
Ratio,  the  Median  Ratio  is  of  course  far  more  convenient  than  the 
Pearson  Coefficient.  Once  a  correlation  table  is  written  out  the 
Median  Ratios  can  be  obtained  with  very  little  computation  or  eye 
strain.  Inspection  of  the  correlation  table  will  tell  about  what  they 
will  be  and  only  a  few  of  the  ratios  will  need  to  be  ranged  in  order. 
I  append  a  sample  calculation  (Fig.  8). 

First  one  makes  an  exact  median  sectioning  of  the  #'s  and  the 
y  's  and  then  counts  the  cases  that  give  negative  ratios. 

By  inspection  one  then  chooses  for  the  y/x  ratios  an  approximate 
median  (here  of  about  .25)  and  for  convenience  draws  a  line  to 
include  these  cases  and  counts  them.  One  then  increases  their 
number  by  adding  the  cases  of  the  next  smallest  ratios  not  included 
or  by  taking  away  the  cases  of  the  largest  ratios  included  until  one 
reaches  the  Median  Ratio  (here  .333).  One  then  repeats  the  process 
oi:  guessing  at  an  approximate  median  for  the  y/x  ratios  and  cor- 
recting it, 

In  making  comparisons  on  the  basis  of  the  median  ratios  we 
must  of  course  bear  in  mind  the  variabilities  of  our  A,  B,  C  and  D. 
In  the  Pearson  Coefficients  the  series  concerned  are  reduced  to  an 
equivalence  in  variability  in  the  process  of  calculation.  With  the 
Median  Ratios,  if  we  wish  to  make  this  reduction  to  terms  of  the 
variability  as  a  unit  we  must  do  it  as  a  separate  operation.  For 
instance  let  A,  B,  C  and  D  be  series  with  variabilities  1,  2,  4  and  5. 
If  then  the  Median  Ratios  found  are 

B/A  =  1.00,    A/B  =  .25,    D/C  =  .625    and    <7/D  =  .40, 
the  Median  Ratios  that  would  be  found  if  the  differences  in  varia- 
bility were  eliminated  would  be  B/A  -=-2/1,  A/B-+-1/2,  etc.,  that 
is  .50,  .50,  .50  and  .50.     If  we  wish  to  compare  the  mutual  implica- 
tion of  A  and  B  with  the  mutual  implication  of  C  and  D  we  must 

go  further  still  and  combine  the  median  — -  •     --'-   -  with  the  median 


28 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


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B_    var.  A    ,    A  var.  B 
still  by  taking   A  '  var.  B        B  var.  A 


MEASUREMENTS    OF   RELATIONSHIPS  29 

Comparison  is  thus  more  awkward  with  the  Median  Ratios  than 
with  the  Pearson  Coefficients,  because  the  latter  method  automat- 
ically both  divides  through  by  the  variability  and  gets  a  measure  of 
mutual  implication.  The  superiority  of  the  Pearson  Coefficients  is 
to  some  extent  specious  for  it  makes  comparison  easy  not  by  re- 
moving difficulties  but  by  presupposing  that  they  do  not  exist.  The 
obvious  additional  steps  needed  in  the  case  of  comparison  of  Median 
Ratios  witness  and  emphasize  the  hypotheses  on  the  basis  of  which 
we  do  compare.  They  may  also  prevent  us  from  inadequate  com- 
parison. For  instance  from  the  facts  that  the  Pearson  r  for  adult 
brother's  longevity  with  adult  brother's  longevity  is  .2853  and  that 
the  Pearson  r  for  stature  with  left  middle  finger  length  is  .6608,  we 
have  no  right  to  conclude  that  the  latter  relationship  is  2.3  times 
a.s  close.  Any  one  who  will  study  the  individual  relationships  in 
these  two  cases1  will  see  that  no  single  ratio  can  express  the  com- 
parison of  the  two  relationships. 

Speed  of  Calculation. 

Onee  the  correlation-table  is  written  out  the  Median  Ratio  can  be 
calcinated  iii  from  one  tenth  to  one  hundredth  of  the  time  taken  for 
the  Pearson  Coefficient. 

Divergence  of  Results  Obtained  from  a  Partial  Sampling  from  the 

Results  from  the  Entire  Series  Sampled. 

The  Pearson  Coefficient  is  for  normal  correlation  by  the  theory 
of  error  the  more  reliable.  Whether  in  the  actual  cases  of  relation- 
ship with  which  we  work,  where  the  distributions  and  correlations 
are  not  exactly  normal  and  where  the  theory  of  error  does  not  apply 
without  modification,  it  is  more  reliable,  is  a  matter  to  be  determined. 
Its  use  of  the  exact  amount  of  every  case  of  the  relationships  makes 
for  superior  reliability,  but  its  weighting  of  extreme  cases  may  some- 
what conterbalance  this. 

The  reason  given  by  Professor  Pearson  for  replacing  Galton's 
method  of  obtaining  the  Median  Ratio  by  this  product-moment 
method  was  this  superior  reliability.  No  other  reason  has  so  far 
as  I  am  aware  ever  been  advanced.  It  is  doubtful  if  Professor 
Pearson  now  would  lay  so  much  stress  on  greater  reliability  in  the 
case  of  normal  correlation  of  normal  distributions,  since  he  has  so 
emphatically  shown  the  rarity  of  both  of  these,  and  has  been  at 
some  pains  to  test  empirically  certain  measures  which  are  valid  re- 
gardless of  the  normality  of  distribution  of  the  two  facts. 

Since  in  almost  every  other  respect  the  Median  Ratio  is  a  more 
advantageous  measure,  it  seems  worth  while  to  determine  empir- 
ically, for  some  typical  relationships,  the  comparative  freedom  from 

1  See  Biometrika,  Vol.  I.,  p.  84,  and  Vol.  I.,  p.  2 1C. 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


H. 


—27  etc. 


TABLE    XII. 


_5  _3  _i  -j-i  +3  4-5 


+27 


—27 

1 

1 

-25 

-23 

1 

1 

2 

1 

1 
1                  1 

1 
1    1 

1 

1      1 

1 

1 

1 

1 

1    1    2 

1 

1 

1 

!8 

1 

1    2   2 

2 

2 

2 

1 

1 

1 

3    2 

1 

21 

••S 

1                         1 

1    2    2 

2 

3 

1      3 

2 

1 

1     1 

432 

2 

32 

I 

1         1 

243 

4 

6 

5     6 

5 

5 

3    3 

1    1 

51 

f 

1         2    2 

223 

7 

5 

6     6 

6 

5 

5    4 

2    1 

1 

61 

2        3 

345 

9 

9 

11    10 

9 

9 

6    5 

3    3 

92 

—  5 

3    3 

366 

10 

10 

11    12 

10 

9 

6    7 

4   4 

1    1        1 

108 

—  3 

1        2        3 

286 

18 

12 

15    15 

14 

13 

9    9 

2    2 

1         1 

129 

-  1 

2   2 

379 

13 

14 

15    15 

15 

15 

9    9 

3    3 

11             11 

139 

+  1 

113 

365 

9 

11 

15    15 

16 

15 

13  13 

763 

3    2    1 

148 

+  3 

1        12 

245 

8 

8 

15    13 

14 

15 

13  12 

552 

2    1    1 

129 

+  5 

1        2 

223 

6 

7 

9     9 

12 

11 

11  10 

653 

2   2    1 

104 

1 

232 

5 

6 

9     9 

11 

11 

9  10 

562 

211             1 

96 

2    2 

3 

4 

6     6 

6 

6 

5     4 

2    1    2 

2    1         1 

53 

1    2 

3 

2 

5     5 

6 

5 

5     4 

1    1    2 

211 

46 

1    1    2 

1 

1      2 

2 

2 

2     2 

761 

1    1 

32 

1    1 

1 

1 

1 

1 

2     2 

441 

2         1 

22 

1 

1 

1 

232 

2 

12 

1 

1 

1 

1         1 

1 

6 

1 

1 

1 

1 

1 

3 

+  27 

1 

1 

1      2   2   81026285862   97103128129  132125102  986456282611    72211 


TABLE  XIII. 


-11  -10  —9  —8  —7  —6  —5  —4  —3  —2  —1 
1 

1   1 
1  1 


+1  +2  +3  +4  +5  4-6  +7  +8  +9  +10+11 


—5 
—4 
-3 
—2 
— 1 

+1 
+2 
+3 
+4 
+5 


1 

1   1   2 
1     2     4 
1  3  5 
1  3  2 
1     3 
1     1  2 

2  1 
2  1 
333 
5  12  2 
8  17  19 
5  20  30 
3  18  20 

1 
2 
1 

4 

17 
20 
24 

1 

5 
11 
17 
20 

1 

1 
6 
8 
12 

1 

4 
4 

11 

1 
1 
3 
5 

1 
1 

1 

2  11  16 

18 

28 

21 

17 

10 

3 

1 

1 

1   4  11 

15 

18 

30 

'20 

1C, 

6 

1 

1 

1   1  4 

11 

13 

17 

'24 

'24 

17 

3 

1 

2 

1  2 

3 

11 

16 

U 

21 

30 

5 

1 

1   1 

1 

2 

3 

3 

10 

11 

19 

18 

9 

7 

1   2 

1 

2 

5 

7 

6 

9 

4 

5 

1 

2 

4 

2 

4 

4 

4  2 

2 

2 

1 

2 

3 

1 

1 

2 

1 

1 

1 

1 

2      2      7    11    20 


1    1 
1 

90110120130     130120110    90    30    20    11      7      2 


11 

18 
39 
89 
112 
118 
128 

124 
118 
107 
86 
39 
23 
11 

1  7 
2 
2 
1 


MEASUREMENTS   OF   RELATIONSHIPS  31 

chance  error  of  it  and  of  the  Pearson  Coefficient.  I  have  also  tested 
the  influence  of  the  number  of  cases  on  the  per  cent,  of  unlike- 
signed  pairs  (which  I  have  called^  U)  because  at  least  for  pre- 
liminary investigations  of  mental  and  social  relationships  the 
formula^  r  =  cosine  -n-U  (where  U  =  the  per  cent,  of  unlike-signed 
pairs,  deviations  being  calculated  from  an  exact  median  sectioning, 
with  no  zero  deviations)  will  often  possess  great  advantages. 

The  accuracy  with  which  the  Pearson  r,  the  Median  Ratio  and 
the  cosine  vU  calculated  from  a  random  sampling  of  a  series  of 
individual  relationships  approximate  the  true  r,  the  true  Median 
Ratio,  and  the  true  cosine  nil  of  the  entire  series  was  experimentally 
determined  in  the  case  of  the  series  A,  B  and  C  (shown  in  Tables 
IX.,  XII.  and  XIII.).1  These  reliabilities  could,  I  suppose,  be 
calculated  by  theory  for  any  given  series  of  relationships  but  it 
seemed  wise  to  determine  them  also  by  experiments  with  real  cases. 
In  calculating  the  results  for  each  draw  of  200,  100  or  of  50  cases 
the  deviations  were  reckoned  always  from  the  true  central  tend- 
encies of  the  total  series,  not  from  the  obtained  central  tendencies  of 
the  draw  itself.  This  saves  much  time  and  introduces  no  error 
relevant  to  the  problem.  The  Median  Ratio  was  taken  simply  as 
the  observed  ratio  of  which  it  was  a  case.  That  is,  if  the  distribu- 
tion of  ratios  was : 

Less  than  1.00  —  49 

1.00  —  12 

over  1.00  —  39, 

the  Median  Ratio  would  be  taken  as  1.00.  If  one  took  as  the  Median 
Ratio  the  average  of  this  observed  ratio  and  the  ratio  halfway  be- 
tween the  40  and  60  percentiles,  the  divergences  for  the  Median 
Ratio  would  be  reduced.  The  results  are  given  in  Table  XIV.  In 
every  case  the  Median  Ratio  means  the  median  of  all  the  ratios 
(y/x  and  x/y),  the  two  series  being  reduced  to  an  equivalence  in 
variability. 

The  relationships  as  calculated  from  the  entire  series  are: 

Series  A  Series  B  [  Series  C 

Pearson  Coefficient  .51  .27  .73 

Median  Ratio  .60  .33  .83 

Cosine  vU  .61  .30  .79 

It  is  clear  from  Table  XIV.  that  if  A7  is  as  great  as  100,  there  is 
no  great  loss  in  precision  from  the  use  of  the  Median  Ratio  method 
or  even  of  the  unlike-signed  pairs  method. 

1  Table  IX.  is  on  page  19. 


I 


32  EMPIRICAL    STUDIES    OF    MEASUREMENT 

TABLE    XIV. 
AVERAGE  DIVERGENCE  OF  OBTAINED  FROM  TRUE  MEASURE  OF  RELATIONSHIP1 

(Figures  in  parentheses  give  the  ranks  of  the  three  methods  in  freedom 
from  chance  error.) 

No.  of  No.  of                    Pearson  Median  Ratio 

Trials  Cases  Coefficient  (Double-entry)  Cosine  -nil 
Series  A 

10  200  .039(1)  .053(2)  .058(3) 

10  100  .065(2)  .062(1)  .101(3) 

10  50  .100  (1)  .155  (3)  .135  (2) 
Series  B 

5  200  .064(2)  .063(1)  .082(3) 

5  100  .105(3)  .072(1)  .075(2) 

10  50  .153(1)  .192(2)  .197(3) 
Series  C 

3  200  .044  (2)  .072  (3)  .013  (1) 

3  100  .032(1-2)  .050(3)  .032(1-2) 

5  50  .119  (2)  .120  (3)  .077  (1) 

The  Advantages  of  Certain  Other  Measures. 

The  Average  Ratio  has  no  advantage  over  the  Median  Ratio  and 
suffers  from  the  disadvantage  of  taking  an  enormous  amount  of 
time  and  being  influenced  so  much  by  extreme  ratios.  No  experi- 
enced worker  with  relationships  would  favor  its  use. 

The  Modal  Ratio  is  in  some  respects  the  most  important  single 
feature  of  the  entire  series  of  relationships,  and  is  probably  a  better 
basis  of  comparison  between  different  relationships  when  either  is 
not  normally  distributed  than  the  Pearson  Coefficient  or  the  Median 
Ratio.  The  observed  Modal  Ratio  from  a  small  sampling  diverges 
so  much  from  the  true  Modal  Ratio  of  the  total  series,  however,  that^ 
it  can  not  be  well  used  alone  unless  the  number  of  ratios  is  500  or 
more:  The  scale  should  also  be  fine.  The  most  probable  true 
Modal  Ratio  inferred  from  a  large  part  of  the  total  distribution  of 

1  It  is  hardly  worth  while  to  compare  the  empirical  divergences  of  Table 
XIV.  for  the  Pearson  Coefficients  with  the  divergences  to  be  expected  from  the 

.7979(1  —  r2) 
formula  A.D.  true  r-obtained  r  =   — - — -7= ,   for  this  formula,   calculated   for 

'  normal '  correlation,  would  not  be  expected  to  fit  very  closely  any  of  the  three 
sets,  A,  B  and  C,  or  to  fit  C  at  all  closely.    A  certain  interest  does  attach  to  the 

.7979(1  —  r2) 
comparison  from  the  fact  that  the  formula  A.D.  ,rue  r- obtained  r   =  - 


has  also  been  proposed  as  the  valid  one.  So  far  as  my  drawings  go,  the  former 
is  surely  the  better.  They  vary  from  it,  moreover,  with  a  constant  deviation 
toward  a  larger  divergence,  the  divergences  by  theory  being: 

Series  A  Series  B  Series  C 

.042  .053  .027 

.059  .074  .038 

.083  .105  .054 


MEASUREMENTS    OF  -RELATIONSHIPS 


the  relationship  is  a  very  valuable  measure  but  one  the  calculation 
of  which  takes  a  long  time  and  involves  presuppositions  about  the 
form  of  distribution  of  the  relationship. 

In  all  cases  the  investigator  of  a  relationship  should  be  observ- 
ant of  the  form  of  distribution  of  the  individual  relationships  and 
of  their  approximate  mode.  Where  the  correlation  table  shows  any 
marked  eccentricity  in  the  distribution  of  the  relationships  the  ob- 
served modal  relationship  at  least  should  probably  be  stated,  even 
though  the  more  reliable  Median  Ratio  or  Pearson  Coefficient  has 
been  calculated. 

The  correlation  (in  the  sense  of  the  slope  of  the  line  which  the 
Pearson  Coefficient  measures)  may  be  inferred  from  the  frequencies 
of  certain  types  of  pairs,  as  in  the  case,  r  =  cos.  irl]  (U  equalling 
the  percentage  of  unlike-signed  pairs  with  median  sectioning). 

The  methods  of  making  this  inference  are  especially  valuable 
when  we  wish  to  compare  two  relationships,  one  (or  both)  of  which 
is  measured  very  crudely,  for  instance,  the  relation  between  health 
and  cheerfulness  and  the  relation  between  intellect  and  morality. 
From  such  measures  as  the  following : 


g  Much 
g  Little 


Health 

Sickly  Healthy 

150  150 


Inferior 


Intellect 
Dull  Bright 

315  285 


250 


450 


1  2  Superior  145 


2G5 


of 


one    can    not    compare    directly    the    closeness    of    relationship 
health  and  cheerfulness  with  that  of  intellect  and  morality. 

The  following  formulas,  suggested  by  Pearson,  are  probably  the 
best  available  for  dealing  with  such  casesT    In  all  N=  the  total 


FIG.  9. 

number  of  pairs;  a,  b,  c  and  d  mean  respectively  the  numbers  of 
^Wi,  £22/i»  x\y-t  and  x2y2  pairs  where  Xj.  means  measures  above  any 
given  degree  of  x  and  x2,  measures  below  it,  and  similarly  for  y1  and 
3/8  (see  Fig.  9). 


34  EMPIRICAL    STUDIES    OF    MEASUREMENT 


,     TT        1  labcdN 

I.     r  =  sin  -  where     F  =  — -, j-^ 


1  H 

cases  being  so  chosen  that  ad  >  &c. 

III.   r  =  sin  *  -^L     — -z^. 
+  l/6c 


t2  -  3),  etc. 
Since 


and 

(a  +  &)  —  (c  +  d) 

-IT" 

7i  and  A;  are  found  from  tables  of  the  probability  integral,  a,  &,  c 
and  eZ  being  known. 

H  is  taken  as  -4=  e~y^ 


H  and  K  are  thus  found  from  tables. 

Of  these^formulas  IV.  is  for  'normal'  correlation  the  most  ac- 
curate. It  presupposes  'normal'  correlation:  I.,  TT  anH  TTT  Hn  not 

When  the  facts  to  be  related  are  measured  on  a  fine  scale  but  in 
terms  of  relative  position  only,  not  of  amount,  the  relationship  may 
be  measured,  as  Spearman  has  shown,  by  the  degree  of  conformity 
of  the  second  member's  position  to  that  of  the  first  member.1  This 
method  suffers  from  the  disadvantage  of  giving  results  only  with 
much  difficulty  comparable  with  other  methods  and  of  taking  much 
more  time  without  being  much  more  reliable  than  the  cosine  irU 
method. 

From  the  reduction  in  variability  of  an  array  of  y  related  to  a 
given  value  of  x  below  the  variability  of  the  total  series  of  y,  the 
correlation  may  be  inferred  on  the  supposition  that  the  correlation  is 
'normal'  and  that  the  variabilities  of  all  arrays  of  y  are  equal. 

The  infrequency  of  'normal'  correlation  and  the  fact  that,  as 

1  See  American  Journal  of  Psychology,  Vol.  XV.,  p.  86  ff. 


MEASUREMENTS    OF   RELATIONSHIPS  35 

shown  in  §  4,  the  variabilities  of  all  arrays  of  y  are  usually  not  equal 
make  this  method  of  no  great  practical  service  except  for  the  few 
cases  where  no  better  method  can  be  used.  l 

Section  4  tested  the  hypothesis  of  equal  variability  of  all  arrays 
of'  y  and  found  it  true  in  some  cases  and  false  for  others.  It  is  some- 
what extraordinary  that  Professor  Pearson  should  in  support  of  his 
coefficient  of  variability  argue  that  the  gross  variability  depends 
on  the  size  of  the  mean  from  which  the  variability  is  measured,  be- 
ing proportioned  to  it,  and  yet  not  recognize  that,  since  the  means 
of  the  arrays  of  y  in  positive  correlation  would  then  increase  as  we 
pass  from  arrays  related  to  low  values  of  x  to  arrays  related  to  high 
values  of  x,  the  variability  of  one  of  the  latter  arrays  should  be 
greater  than  that  of  one  of  the  former. 

§8.     The  Attenuation  of  Measurements  of  Relationship 

Chance  inaccuramps  in  flip  m-lonnql  measures  make  the  relation- 
ship obtained  therefrom  vary  toward  zero  from  the  relationship  that 
would  be  found  with  accurate  measures.  C.  Spearman  announced 
in  the  American  Journal  of  Psychology,  Vol.  XV.,  pp.  89-91,  that 
the  following  formulas  gave  the  necessary  correction  ; 


a) 


rq,q, 


where  rp,q,=  ihe  mean  of  the  correlations  between  each  series  of 

values  obtained  for  p  with  each  series  obtained  for  q  ; 

»yy=the  average  correlation  between  one  and  another  of 

these  several  independently  obtained  series  of  values 

of  p; 

rgV=the  same  as  regards  q; 

and  rp<,=  the  required  real  correlation  between  the  true  objective 
values  of  p  and  q. 


where  m  and  n  =  the  number  of  independent  gradings  for  p  and  q 
respectively  ; 

1  Cases,  that  is,  where  we  know  the  variability  of  a  related  array  but  lack 
the  data  needed  for  the  use  of  the  better  methods.  For  instance,  we  may  find 
the  variability  of  100  men  eminent  in  engineering  science  in  early  liking  for 
arithmetic  to  be  only  30  per  cent,  as  great  as  the  variability  of  men  in  general 
and  so  infer  the  amount  of  relationship  between  early  liking  of  arithmetic  and 
engineering  ability.  The  actual  rating  of  a  random  sampling  of  men  in  both 
early  liking  for  arithmetic  and  engineering  ability  would  be  hardly  possible. 


36  EMPIRICAL    STUDIES    OF   MEASUREMENT 

ry^  — the  mean  correlation  between  the  various  grad- 

ings  for  p  and  those  for  q ; 
and  rp,,g,,  =  the  correlation  of  the  amalgamated  series  for  p 

with  the  amalgamated  series  for  q. 

He  has  been  criticized  with  some  venom  bv  Karl  Pearson 
(Biometrika,  Vol.  III.,  p.  160),  who  believes  these  formulas  wrong, 
and  concludes  that  "Perhaps  the  best  thing  at  present  would  be 
for  Mr.  Spearman  to  write  a  paper  giving  algebraical  proofs  of  all 
the  formulas  he  has  used,  and  if  he  did  not  discover  their  erroneous 
character  in  the  process,  he  would  at  least  provide  tangible  material 
for  definite  criticism,  which  it  is  difficult  to  apply  to  mere  unproven 
assertions. ' ' 

These  formulas  of  "Spearman's,  if  correct,  are  of  importance. 
They  should  be  proved  valid  or  replaced  by  formulas  that  are  valid. 
The  first  formula  may  be  replaced  by 


*  —  ovp,2 1/oy2  —  vtq? 

where  rpq  and  rp>q>  are  as  above  and 

%/  =  the  mean  square  deviation  of  the  series  of  measures  of  p ; 

oy  =  the  mean  square  deviation  of  the  series  of  measures  of  q ; 

o-q,,  =  the  mean  square  deviation  of  the  different  measures  of  p  in 

the  same  individuals ; 

ov  =  the  mean  square  deviation  of  the  different  measures  of  q  in 
the  same  indivduals.1 

The  presupposition  of  this  formula  and  of  Spearman's  first 
formula  is.  that  the  attenuation  is  due  to  chance  errors.  Dr.  Clark 
Wissler  has  called  attention  to  the  fact  that,  where  practise,  fatigue 
and  other  constant  influences  help  to  cause  the  different  observations 
of  a  fact  to  vary,  these  formulas  will,  therefore,  pive  inaccurate 
results.2  ^)  ^  /3  •> 

Of  these  two  formulas,  Spearman's  possesses  the  advantage  of 
being  usable  in  cases ^Fere  the  twcTtraits  are  not  measured  in  units 
f  5 )  o£  amount,  such  as  allow  the  variabilities  of  the  two  traits  to  be 
calculated;  the  formula  of  Boas  has  the  advantage  of  being  more, 
rapid  and  convenient  in  cases  where  the  variabilities  of  the  two 
traits  can  be  calculated. 

No  active  attention  has  so  far  as  the  writer  knows  been  yet  given 
^  to  formula  (2)  above.3    Practical  necessity  seems  to  justify  the  labor 

1  This  formula  is  due  to  Professor  Franz  Boas.     See  also  the  note  by  Dr.  C. 
Wissler  in  Science,  Vol.  XXII.,  p.  309  ff. 

2  Loc.  cit.  in  note  1. 

*  Spearman's  second  formula  has  the  advantage  of  measuring  the  probable 
true  correlation  by  the  actual  changes  produced  in  the  obtained  '  raw  '  correlation 
by  a  certain  increase  in  accuracy.  The  nature  and  validity  of  the  presupposi- 
tions upon  which  it  is  based  I  am  not  competent  to  discuss. 


MEASUREMENTS   OF   RELATIONSHIPS  37 

of  testing  it  (and  in  a  measure  the  first  formula  also)  inductively. 
This  I  have  done  to  some  extent  for  values  of  r  where  the  r's  from 
accurate  measures  are  from  .70  to  .80  in  connection  with  my  'Meas- 
urements of  Twins'  (Archives  of  Philosophy,  Psychology  and  Scien- 
tific Methods,  No.  1,  September,  1905). 

I  had  records  from  50  pairs  of  twins  in  5  tests  of  efficiency  of 
perception;  (1)  in  marking  A's  on  a  sheet  of  printed  capitals,  (2) 
in  marking  A's  on  a  second  sheet  of  printed  capitals,  (3)  in  mark- 
ing words  containing  e  and  r  on  a  page  of  Spanish,  (4)  in  marking 
words  containing  a  and  t  on  a  page  of  Spanish  and  (5)  in  marking 
misspelled  words  on  a  page  of  narrative,  100  of  whose  words  were 
misspelled.  I  had  also  6  tests  in  efficiency  of  controlled  association, 
tests  6  and  7  being  addition,  8  and  9  being  multiplication  and  10 
and  11  being  writing  the  opposites  of  two  lists  of  words. 

If  we  combine  all  5  of  the  tests  of  efficiency  of  perception  allow- 
ing approximately  equal  weight  to  each,  we  have  a  measure  which 
is  presumably  close  to  the  true  measure  of  a  child's  capacity  at  a 
certain  day  and  hour  to  pick  out  small  details  efficiently.  The  cor- 
relation between  twin  and  twin  is  for  this  combined  score  .697. 
Similarly  the  combined  measure  for  addition,  multiplication  and 
opposites  gives  a  measure  presumably  close  to  the  true  measure  of  a 
child's  ability  at  a  certain  day  and  hour  to  make  proper  mental 
connections.  The  correlation  between  twin  and  twin  is  .815.  The 
.697  and  .815  are  presumably  only  slightly  below  the  true  r's. 

Now  the  correlations  for  twin  and  twin  in  tests  1-11  were  in 
order  .607,  .633,  .595,  .428,  .754,  .645,  .644,  .653,  .579,  .734  and  .560. 
Subjecting  these  values  to  correction  by  Spearman's  formulas, 
taking,  as  he  does,  the  mean  of  both  corrected  r's  I  obtained 
for  the  perception  tests:  Marking  A's,  true  r=.69;  marking 
letters  in  words,  true  r  =  .71 ;  misspelled  words,  not  corrected 
because  only  one  test  was  given.  The  Spearman  correction 
thus  produced  results  in  accord  with  the  expectation  derived  from 
the  value  r— .697  for  the  combined  mark.  For  the  association  tests 
J  obtained  after  correction:  Addition,  true  r=.75;  multiplication, 
true  r  =  .84 ;  opposites,  true  r  =  .90.  The  average  of  these,  .83,  is 
again  closely  in  accord  with  the  .815  from  the  combined  measure. 
In  both  cases  the  result  by  correction  is  slightly  higher  than  the 
result  empirically  obtained  from  the  more  accurate  data,  as  of 
course  it  should  be. 

I  have  made  a  test  ad  hoc  in  the  case  of  a  series  of  100  pairs 
drawn  at  random  from  Series  B  which  give  a  true  r  of  .281.  These 
100  pairs  of  accurate  measures  I  made  inaccurate  artificially.  I 
then  calculated  the  r's  obtained  from  such  inaccurate  measures, 
applied  the  Spearman  formulas  and  in  so  far  tested  their  validity. 


38  EMPIRICAL    STUDIES    OF    MEASUREMENT 

Special  precautions  were  taken  to  have  the  errors  artificially  in- 
duced in  the  200  measures  such  as  would  come  in  reality  from 
variable  errors  of  apparatus,  observation  and  record.  The  errors 
were  in  fact  a  random  sampling  of  the  errors  actually  made  by  a 
psychologist  in  estimating  areas.  A  series  of  121  rectangles  of 
approximately  the  same  shape,  40,  41,  42  ...  160  sq.  cm.  with 
also  many  duplicates  were  used.  The  area  of  each  was  estimated, 
the  slips  being  drawn  in  a  random  order,  and  the  error  -\-  or  — 
from  the  true  area  was  recorded.  The  errors  used  by  me  were 
those  made  after  from  3  to  5  trials  with  the  series  and  were  little  in- 
fluenced by  practice  (the  sums  of  the  errors  regardless  of  signs  were 
for  successive  repetitions  of  the  series  605,  614,  563,  613,  587,  637, 
531,  542,  578,  581).  I  used  the  deviation  from  the  standard  if  the 
constant  error  for  the  given  area  was  less  than  1  sq.  cm.  and  the 
approximate  deviation  from  the  subject's  own  average  judgment 
if  the  constant  error  was  over  1  sq.  cm.  The  errors  taken  were 
those  (10  in  each  case)  made  with  areas  43  sq.  cm.  up  through  122 
sq.  cm.,  four  errors  being  taken  for  each  of  the  200  accurate  meas- 
ures. These  errors  were  assigned  to  the  accurate  measures  so  that 
the  magnitude  of  the  area  with  which  the  error  of  estimation  was 
made  corresponded  roughly  to  the  magnitude  of  the  measure  to 
which  the  error  was  assigned.  Thus  errors  from  areas  43-53  would 
be  put  with  measures  —  27,  —  25,  —  23  and  the  like,  and  errors  from 
areas  110-122  would  be  put  with  measures  +17,  -f- 19,  -(-27  and 
the  like.  The  true  measures  and  the  errors  assigned  to  each  are 
given  in  Table  XV. 

If  now  to  each  true  measure  is  added  (regarding  signs)  its  as- 
signed error,  we  have  (four  errors  having  been  assigned  to  each) 
four  series  of  inaccurate  measures  of  two  series  whose  true  values 
and  true  correlation  are  known.  These  facts  give  the  data  for  test- 
ing the  Spearman  formulas.1 

1  These  errors  can  of  course  be  used  with  any  series  of  400  or  less  measures 
to  test  Spearman's  formulae,  as  I  have  done  for  this  series  (r  =  .281  of  Series  B) . 


MEASUREMENTS    OF    RELATIONSHIPS 


39 


TABLE    XV. 


True  i 
a       —19 
b       —17 
e       —15 
d      —15 
etc.  —15 
—15 
—13 
—13 
—13 
—11 
—11 
—11 

—  3 
—  2 

+  9 
+  6 

0 

—  8 
+   1 
—  8 
+   1 
+   1 
—  2 
0 

Errors  Assigned 
+  2      —  4 
+   5      —  1 
+  4            0 
—  2      —  2 
-  1       -  1 
+   1      +2 
+  6      —  2 
+   1       -  1 
+  2      +1 
—  5      +11 
0—6 
—  2            0 

—  2 
-  4 
+   1 
—  3 
—  3 
+  2 
—  3 
+  4 
+  3 
—  4 
+  4 
+  7 

Truey 
a      —11 
b         -   1 
c        —27 
d       —  9 
etc.  +  3 
+  7 
—11 
—  3 
+  13 
—13 
—  5 
—  5 

o 

+  12 
+  7 
0 
+  7 
+  3 
+  6 
+  4 
+   1 
—  4 
+  7 
+  7 

Errors  Assigned 
_  4      —  4       +5 

—13      —  9       —  7 
+  2       —  4      —  3 
+  3—8            0 
+  7      -  2      —  3 
+  10     —  7      +7 
-5+9            0 
+  3      +3      +6 
-  1      —  1      -  6 
-  3      +7      —  5 
-  4      +2      +4 
—  1           0—6 

—11 

—12 

— 

1 

0 

—  3 

—  3 

-  1 

—  6 

—  8 

—  9 

—  9 

Q 

+ 

7 

+ 

9 

0 

—  5 

+  3 

0 

0 

+  9 

—  9 

+  13 

+ 

8 

+ 

3 

—12 

—  3 

+  1 

—  1 

+  1 

+  6 

—  9 

+  2 

+ 

1 

— 

3 

+  2 

—  1 

Q 

+  6 

—  5 

—  3 

—  9 

—  4 

— 

1 

— 

4 

—  3 

-  1 

—  5 

+  6 

+  3 

—  6 

—  9 

—  6 

+ 

6 

+ 

4 

+  12 

+  5 

o 

+  3 

—  6 

+  2 

—  9 

—  3 

— 

2 

— 

2 

0 

+  13 

—  8 

-  1 

+  6 

+  4 

—  7 

0 

+ 

2 

— 

2 

+  5 

-  1 

+  5 

—  5 

+  1 

—  5 

-  7 

—  2 

— 

6 

+ 

3 

+  2 

+  7 

+   1 

+  2 

—  4 

+  3 

—  7 

+  7 

+ 

4 

+ 

5 

—  2 

+  9 

+  2 

.     K 

—  3 

K 

—  5 

—  6 

— 

2 

— 

2 

+  3 

—13 

+   1 

+   1 

+  4 

+  6 

—  5 

+  13 

— 

4 

— 

4 

+  3 

—  9 

—  6 

0 

+  3 

—  3 

—  5 

+  8 

— 

3 

— 

7 

0 

—  7 

—  6 

+  2 

—  2 

+  2 

—  5 

—  3 

+ 

9 

— 

4 

—  3 

—  7 

+   1 

+  3 

—  5 

+  2 

—  5 

—  7 

— 

3 

— 

5 

+  2 

—  3 

o 

+  6 

—  2 

—  7 

—  5 

+  7 

+ 

2 

0 

1 

—  3 

+  7 

+   1 

+  4 

—  2 

—  5 

-  1 

+ 

4 

0 

—  2 

—  3 

+  5 

+  10 

—  5 

+  4 

—  5 

+  4 

— 

2 

+ 

4 

—  3 

—  3 

—14 

+  6 

—  3 

—  5 

—  5 

+   1 

— 

3 

— 

3 

—  3 

3 

-  1 

—11 

—  3 

—  2 

—  5 

+  6 

+ 

3 

— 

3 

—  9 

+  13 

—  3 

—  4 

+  3 

+  11 

—  3 

+  3 

+ 

7 

+ 

6 

—  5 

—11 

+  11 

+  7 

O 

—11 

—  3 

+  7 

+ 

3 

+ 

6 

—  2 

—  9 

+  5 

—  2 

—  3 

+  3 

—  3 

-  1 

— 

7 

— 

2 

—  7 

—  9 

—  6 

—  3 

—  5 

+  5 

—  3 

+  3 

— 

9 

+ 

6 

—  2 

<T 

+  7 

—  4 

+  8 

+  7 

—  3 

+  3 

— 

4 

+ 

3 

—  3 

—  5 

+  4 

+  4 

-  1 

+  3 

—  3 

0 

+ 

2 

+ 

5 

0 

—  5 

+  6 

—  5 

—  4 

+  1 

—  3 

-  1 

—12      — 

5 

+  6 

—  5 

+  5 

—  9 

+  5 

—  3 

-  3 

—10 

+ 

2 

+ 

9 

+  2 

—  5 

0 

0 

—  5 

+  3 

—  3 

+  1 

— 

5 

+ 

4 

—  5 

-  1 

0 

+  4 

1 

—  3 

-  3 

+  1 

+ 

1 

0 

—  8 

+   1 

—11 

—  5 

+  12 

+  7 

—  3 

+  5 

+ 

5 

— 

6 

0 

+   1 

—  2 

—12 

—10 

+  6 

—  3 

0 

— 

3 

+ 

3 

0 

+   1 

+  13 

—  2 

+  6 

O 

—  3 

+  3 

+ 

5 

-f- 

3 

-  1 

+  5 

—  4 

+  11 

—  1 

—  6 

—  3 

—11 

+ 

6 

+ 

4 

—  9 

+  9 

+  5 

+  7 

—  1 

—11 

-  1 

+  2 

— 

6 

+  13 

1 

—  9 

—  6 

—  2 

-  1 

-  1 

—  1 

—  4 

+ 

8 

+ 

7 

+  4 

—  5 

—  2 

+  4 

—  7 

+   1 

-  1 

+   1 

— 

8 

— 

4 

fj 

—  3 

+'  4 

+  7 

—  2 

0 

—  1 

+  2 

+ 

8 

— 

2 

+  2 

—  3 

—  1 

0 

+  12 

+11 

40 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


ue  x 

fa 

^           TABLE    XV.   (continued) 

flJ>             AC-           X4                                                J* 

Errors  Assigned                                   True  y 

Errors  Assigned 

—   1 

—  9 

+   9 

—  1 

0 

+   1 

—  5 

+  5 

+  9 

+  3 

-   1 

+  8 

—  5 

+  5 

0 

+  3 

0 

+  1 

+   1 

0 

-  1 

+   1 

—  5 

—  5 

-  1 

+  5 

0 

+   1 

+  2 

—  3 

-   1 

+  2 

+  7 

-  1 

+  4 

+  7 

+  3 

—  2 

+  14 

—  6 

-f- 

+  3 

—  5 

—  6 

0 

+  17 

0 

+  4 

—  1 

—  8 

4- 

—  6 

+  3 

—15 

—  2 

+  11 

—13 

+  7 

+  4 

+   1 

+ 

+   1 

—  2 

+  6 

+   1 

+  11 

+  2 

—  4 

+  4 

+  12 

-j- 

+  6 

+  8 

+  2 

0 

+  5 

+  7 

+  5 

+  4 

—  7 

4. 

+   1 

0 

+  2 

0 

+  1 

—  7 

—10 

—  9 

+  8 

+ 

+   1 

+  3 

+  2 

—  8 

+   1 

+  2 

+  6 

—  9 

—  3 

+ 

+  17 

+   1 

—  9 

2 

+   1 

+  5 

0 

+  3 

+  7 

-f- 

—  3 

+  4 

—  5 

—  6 

—  3 

—  9 

—  1 

+  4 

—  6 

+ 

0 

+  7 

+  2 

+  4 

—  5 

+  3 

+  7 

-  1 

+  6 

+ 

2 

+  1 

+  7 

—  9 

—  7 

+11 

+  3 

—  2 

—  3 

+ 

+  3 

—  6 

+  4 

-  1 

A 

1 

—  7 

—  2 

—  3 

+ 

0 

+  3 

—  5 

—  4 

—  9 

+  3 

—  1 

+  10 

—  8 

+ 

+  5 

+  1 

+  5 

+  4 

—13 

—  6 

+  6 

+  6 

—  1 

+  3 

—  3 

—  9 

—  3 

+  6 

+   1 

—  4 

+  3 

—  3 

0 

+  3 

+   1 

+  3 

+   1 

-  1 

+   1 

0 

—  3 

—  3 

—  2 

+  3 

o 

O 

—  2 

—  2 

—  1 

+  4 

0 

0 

+  7 

+  3 

+  7 

0 

+  3 

+  5 

—  5 

+   1 

2 

+  3 

—  2 

+  5 

+  3 

+  2 

+  4 

-  1 

+  11 

+   1 

-  1 

+  1 

+  2 

+  5 

+  5 

+  3 

+  5 

—11 

+  5 

+  8 

—  2 

—12 

A 

+  5 

+  3 

+  2 

0 

—  9 

+   1 

+  7 

—  3 

+16 

+  4 

+  5 

—  4 

—  8 

—  4 

+  3 

+   1 

-  1 

—  2 

—  2 

—  6 

+  5 

+  10 

+  9 

—  2 

+  3 

—  7 

—  2 

—  7 

+  6 

+  5 

+  5 

—  5 

—  5 

+  4 

+  3 

—  9 

+  1 

—  3 

—  4 

+   1 

+  5 

+  10 

—  6 

—  8 

+  12 

—19 

+  2 

+  5 

+   1 

—  4 

+  7 

—  6 

A 

+  2 

—  4 

+  11 

—  4 

—  3 

—  8 

—  5 

+  7 

+  5 

+  10 

—  5 

—  3 

+  7 

+  7 

+  15 

—  7 

-  7 

+  7 

—  2 

—  7 

+  2 

—  5 

+  5 

—  3 

—  3 

—  3 

+  14 

+  7 

+  7 

+  1 

+  15 

1 

+   1 

+  2 

+   1 

—  4 

—  4 

+  7 

—  8 

+  6 

—  6 

+10 

—  3 

—  2 

+  7 

+  3 

—11 

+  7 

+  8 

<J 

—10 

Q 

—  5 

+  7 

—  2 

—  6 

A 

+  7 

+  7 

+  6 

—  3 

—  4 

—  7 

+  5 

—  8 

—  5 

+  8 

+  9 

—10 

—  3 

+   1 

+  7 

+  9 

0 

—  3 

+  1 

+  5 

+  9 

+  7 

+  15 

—  7 

—  7 

+  7 

+  3 

—  7 

+  6 

+  6 

+  9 

+  3 

—  7 

+  6 

+  6 

+  3 

+  10 

—  3 

0 

—  4 

+  9 

—10 

—  9 

+  6 

—  6 

+  1 

+   1 

—13 

—  2 

—  3 

+  9 

+  7 

+  6 

—  3 

—  4 

—  5 

+  7 

+  6 

+  5 

—13 

+  9 

—10 

—  3 

4-   l 

+  7 

—  7 

-  1 

+   1 

—  6 

+  4 

+  11 

—  1 

0 

0 

+  4 

+  5 

—13 

+  4 

—  6 

+  6 

+13 

+  10 

—  2 

0 

+  11 

+13 

0 

—  6 

+  4 

0 

+  13 

—16 

—  5 

—  9 

+  11 

—11 

+  5 

—11 

0 

+  9 

+  15 

+  3 

—  6 

+   1 

—  5 

+   1 

0 

+  5 

+   1 

0 

+  17 

—  1 

+  3 

—  4 

+  8 

+  13 

-  1 

—  6 

+  2 

0 

+17 

+  2 

+  5 

—  3 

—  4 

+  11 

+  7 

—  9 

-  1 

+  14 

+  17 

—  3 

+  6 

+  5 

—11 

+  1 

0 

+  7 

—  4 

—15 

+  17 

+  3 

—  3 

+  3 

—  1 

+  1 

+  3 

—  2 

+  5 

—  3 

+25 

+  2 

—  1 

+  2 

+  2 

+  7 

—10 

—  9 

+  6 

—  6 

MEASUREMENTS    OF    RELATIONSHIPS 


41 


Let  us  call  the  four  series  of  inaccurate  measures  obtained  with 
the  four  errors,  Xa,  Xb,  Xc,  Xd,  and  Ya,  Yb,  Yc,  Yd. 

Call  the  series  obtained  by  averaging  each  member  of  Xa  with 
its  correspondent  in  Xb,  Xab. 

Let  Xcd,  Yob  and  Ycd  have  similar  meanings. 

Call  the  series  obtained  by  averaging  each  member  of  Xa  with 
the  corresponding  Xb,  Xc  and  Xd,  Xabcd. 

Let  Yabcd  have  a  similar  meaning. 

We  have  then  4  very  inaccurate  measures  of  X  in  every  one  of 
the  100  pairs;  so  also  of  Y.  We  have  two  less  inaccurate  measures 
Or  X  and  also  of  Y  in  each  pair.  We  have  one  still  better  measure, 
the  best  obtainable  from  our  data. 

We  may  then  calculate  the  corrected  r  according  to  Spearman, 
using  many  different  combinations  of  the  r's  obtained  from  the 
above  series.  The  combinations  which  I  have  used  and  the  results 
follow  in  Table  XVI.1 

The  correspondence  of  the  coefficients  corrected  by  Spearman's 
formulas  with  the  actual  coefficient  from  accurate  measures  is  satis- 
factory. 

TABLE    XVI. 
rxowithxi  =.731 


rxawlth,.  =.142 

rx6with,»  =.208 

rxJwithya  =.243 

rt(  the  average 

of  the  four)  =.169 

rxabwltoyab  =.212 

r  xcd  with  ycd  =.221 

Txab  with  ycd  =.239 

Txcdwlthyab  =.170 

r2(  the  average 

of  the  four)  =.2105 

Txabcd  with  yabcd    =.260 

=  .260 


=  .289 


==  =.277 

rxabw.xcdryabv.ycd 


fxab  with  xcd 
Tyab  with  ycd 


=  .803 
=  .717 


t.  c. 

rs 


MEI-i 

Average  by  all  for- 

mulae 
Median  by  all  for- 

mulae 
True  relationship 


§  9.    Minor  Advice  to  Students  of  Mental  and  Social  Relationships 

As  a  rule  nothing  should  be  taken  for  granted  about  any  relation- 
ship and  the  result  of  any  calculation  should  be  to  express,  not  to 
replace,  the  comprehension  of  a  fact  about  the  series  of  individual 
relationships. 

1  In  all  the  calculations  I  have  assumed  the  original  0  as  the  central  tend- 
ency from  which  to  reckon  deviation  values.  To  have  turned  each  of  the  200 
values  of  each  of  the  fourteen  series  into  a  new  deviation  measure  would  have 
added  practically  nothing  to  the  general  result  in  the  way  of  accuracy.  The 
labor  of  2,800  little  sums  in  addition  and  2,800  copyings  of  numbers  could  be 
more  profitably  spent.  My  figures  are  on  this  basis. 


42 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


Measurements  should  be  on  the  finest  scale  that  can  be  recorded 
without  special  difficulty.  The  attenuation  by  chance  error  is  thus 
diminished  and  the  time  taken  in  making  a  more  elaborate  corre- 
lation table  can  be  saved  ten  times  over  by  the  use  of  the  Median 
Ratio. 

The  central  tendency  from  which  one  measures  deviations  should 
be  chosen  with  care  so  that  it  stands  for  some  reality  divergence 
from  which  is  significant. 

In  the  relationship  given  in  Table  XVII.,1  for  instance,  from 
what  point  should  one  reckon  deviations?  The  authors  take  the 
mean,  56.568.  But  there  is  much  to  be  said  for  taking  the  modal 
adult  life  (at  about  70),  since  that  represents  an  important  real 
tendency  and  the  force  of  heredity  in  determining  departures  from 
that  tenctency  is  perhaps  more  important  than  its  form  in  deter- 
mining departures  fromthe  rather  arbitrary  age,  56.5^8.  The  re- 


TABLE    XVII. 


23    28    33    38    43    48    53      58     63 


73    78    83    88    93    98 


Totals 

of 
Arrays 


23 

10 

20 

8 

14 

9 

8 

5 

104 

4 

15 

11 

6 

7 

2 

133 

28 

20 

18 

15 

6 

9 

13 

8 

43 

7 

5 

6 

1 

9 

2 

126 

33 

8 

15 

18 

12 

14 

8 

8 

93 

11 

8 

3 

10 

7 

2      1 

137 

38 

14 

6 

12 

12 

8 

11 

9 

42 

11 

10 

15 

5 

6 

2 

127 

43 

9 

9 

14 

8 

8 

8 

13 

53 

7 

12 

6 

8 

3 

2 

115 

48 

8 

13 

8 

11 

8 

16 

6 

114 

17 

11 

6 

9 

7 

2 

137 

53 

5 

8 

8 

9 

13 

6 

8 

73 

6 

9 

11 

10 

9 

3      1 

116 

58 

10 
4 

4 
3 

9 
3 

4 

2 

5 
3 

11 
4 

7 
3 

53 
3  1 

8 
3 

15 
6 

4 

4 

12 

7 

7 
4 

2   1 
1   1 

107 
52 

63 

4 

7 

11 

11 

7 

17 

6 

83 

16 

18 

22 

11 

10 

3 

154 

68 

15 

5 

8 

10 

12 

11 

9 

156 

18 

28 

31 

19 

12 

9   4 

212 

73 

11 

6 

3 

15 

6 

6 

11 

44 

22 

31 

40 

16 

13 

3   1   1 

193 

78 

6 

1 

10 

5 

8 

9 

10 

127 

11 

19 

16 

28 

17 

12   3   2 

176 

83 

7 

9 

7 

6 

3 

7 

9 

74 

10 

12 

13 

17 

12 

8   3 

134 

88 

2 

2 

2 

2 

2 

2 

3 

21 

3 

9 

3 

12 

8 

8      1 

62 

93 

1  1 

4 

1 

3 

3 

13 

98 

1 

1 

1 

2 

1 

6 

Medians 

of        49    43    47    51    52    55    57      59     62    65    67    68    65    73    74    76 
Arrays 

Pearson  Coefficient  =  .2853,  C.T.  being  56.568. 
Median  Ratio  =  .479,  C.T.  being  59.4. 

1  The  relationship  between  brother  and  brother  in  length  of  life  in  cases  where 
both  brothers  are  21  or  over,  from  '  The  Inheritance  of  the  Duration  of  Life '  by 
M.  Beeton  and  K.  Pearson,  Biometrika,  Vol.  I.,  p.  84.  I  have  divided  the  array 
of  58  so  as  to  make  a  median  sectioning  of  the  series.  In  the  original  the  array 
for  58  is  given  simply  as  14,  7,  12,  6,  8,  15,  10,  12,  11,  21,  8,  19,  11,  3,  2.  I 
have  also  added  approximate  medians  of  arrays. 


MEASUREMENTS    OF    RELATIONSHIPS 


43 


lationship  in  the  latter  case  (70.5  being  taken  as  the  central  tend- 
ency) is  closer,  the  Median  Ratio  being  .54  or  about  7  higher  than 
the  Median  Ratio  when  divergences  are  calculated  from  56.5.  The 
Modal  Ratio  is  unchanged. 

It  should  be  evident  from  the  facts  stated  in  previous  sections 
that  it  is  out-and-out  folly  to  be  content  with  calculating  for 
every  relationship  studied  the  same  type  of  coefficient.  Nothing 
short  of  the  entire  correlation  table  is  the  adequate  measure  of  the 
relationship  in  question.  Any  measure  of  one  central  tendency  of 
relationship  may  be  misleading,  for  the  relationship  may  be  bimodal. 
When  the  observed  modal  relationship  is  clearly  not  near  the  Pear- 
son Coefficient  the  latter  should  be  accompanied  by  the  former.  So 
also  if  the  modal  relationship  is  clearly  not  near  the  median  rela- 
tionship. 

The  averages  or  medians  or  modes  of  the  arrays  should  be  cal- 


FIG.  10. 


44 


EMPIRICAL    STUDIES    OF    MEASUREMENT 


culated  and  stated,  and  unless  the  relationship  is  uniform  (within 
the  limits  of  chance  error)  throughout  the  course  of  the  series  a  most 
probable  curved  line  to  fit  the  entire  series  should  be  calculated  in- 
stead of  the  slope  of  a  straight  line. 

For  instance,  the  Pearson  Coefficient  for  the  relation  between 
adult  brother  and  adult  brother  in  longevity  is  given  by  Beeton  and 
Pearson  as  .2853.  The  relation  is  sufficiently  close  to  uniformity  for 
all  values  of  x  to  make  a  linear  relation  at  least  approximately  true 
(if  we  consider  also  the  similar  relation  between  sister  and  sister). 
The  relation  is,  however,  by  no  means  identical  with  other  relations 
giving  a  similar  coefficient,  for  the  modal  relationship  is  approxi- 
mately 1.00.  This  can  be  seen  at  a  glance  from  the  graphic  repre- 
sentation of  the  correlation  table  (Fig.  10)  or  the  distribution  of 
the  ratios  (deviations  are  reckoned  from  59.4  and  59.4  as  central 
tendencies)  in  Fig.  11. 

The  .2853  then  does  not  mean  that  the  most  likely  value  of 
B  —  C.T.  of  B  is  near  .2853  X  (A  —  C.T.  of  A),  nor  that  the  forces 
producing  correlation  tend  to  make  B/A  =  .2853,  divergencies  from 


IS-W  40-65   6S-W  W-HS  115-00 

FIG.  11.  Frequencies  of  different  degrees  of  relationship  in  the  case  of  fra- 
ternal longevity.  The  numbers  stand  for  the  ratios  in  per  cents,  the  heights  for 
their  relative  frequency.  The  mode  is  at  very,  very  close  resemblance,  or  ratios 
of  90  to  115  percent. 

this  being  due  to  minor  causes  producing  variations  in  the  correla- 
tion. On  the  contrary  the  .2853  represents  a  most  ambiguous  sum- 
mation of  the  force  of  a  tendency  to  identical  longevity  and  many 
other  forces.  If  the  authors  had  not  given  the  full  correlation 
table,  the  .2853  would  evidently  have  been  definitely  misleading. 

The  determination  of  the  most  likely  law  of  relationship  for  ft 
series  of  pairs  may  then  be  theoretically  and  practically  a  different 
problem  for  eacIT  particular  case,  a  problem  to  solve  which  we  need 
not  only  certain  mathematical  technique  but  also  abundant  knowl- 
edge of  other  similar  relationships  and  of  the  entire  body  of  facts 
relevant  to  the  relationship  in  question.  Thus  the  same  set  of  pairs 
could  properly  be  interpreted  on  the  basis  of  a  linear  relationship 


MEASUREMENTS   OF   RELATIONSHIPS  45 

when  they  were  male  brothers'  first-rib  lengths,  and  could  not  prop- 
erly be  so  interpreted  if  they  were  related  body-strengths  and  earn- 
ings in  dollars.  For  we  have  evidence  from  cephalic  index,  stature 
and  the  like  to  justify  some  expectation  of  linear  correlation  for 
fraternal  relationships  in  features  of  anatomy,  whereas  what  evi- 
dence we  have  concerning  the  relationship  between  body-strength 
and  earning  capacity  in  individuals  goes  to  show  that  it  is  far  less 
close  for  those  of  high  earning  capacity  than  for  those  of  very  low 
earning  capacity. 


-. 

Ill 


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